QUINE, NEW FOUNDATIONS, AND THE

PHILOSOPHY OF SET THEORY

W. V. Quine’s set theory, New Foundations, has often been treated

as an anomaly in the history and philosophy of set theory. In this
book, Sean Morris shows that it is, in fact, well motivated, emer-
ging in a natural way from the early development of set theory.
Morris introduces and explores the notion of set theory as explica-
tion: the view that there is no single correct axiomatization of set
theory but rather various axiomatizations that all serve to explicate
the notion of set and are judged largely according to pragmatic
criteria. Morris also brings out the important interplay between
New Foundations, Quine’s philosophy of set theory, and his philo-
sophy more generally. We see that Quine’s early technical work in
logic foreshadows his later famed naturalism, with his philosophy of
set theory playing a crucial role in his primary philosophical project
of clarifying our conceptual scheme and, specifically, its logical and
mathematical components.

SEAN MORRIS is Associate Professor of Philosophy at the Metropolitan

State University of Denver. He has written extensively on Quine
and the history of analytic philosophy with a particular emphasis
on Quine’s work in the foundations of mathematics.
QUINE, NEW FOUNDATIONS,
AND THE PHILOSOPHY OF
SET THEORY

SEAN MORRIS
Metropolitan State University of Denver
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This book is dedicated to Bill Hart and to Haewon and John.
 Contents

Preface page ix

Introduction 

PART I SET THEORY’S BEGINNINGS 

 Cantor and the Early Development of Set Theory 
. The Beginnings of Set Theory as a Mathematical Discipline 
. From the Potential to the Actual Infinite 
. Cardinals, Ordinals, and the Continuum Problem 

 Cantor, Russell, and Zermelo and the Set-Theoretic Paradoxes 

. Russell and the Discovery of the Paradoxes 
. Avoiding the Paradoxes: Cantor and the Absolute Infinite 
. Resolving the Paradoxes: Zermelo and Russell 

 New Foundations and the Beginnings of Quine’s Philosophy

of Set Theory 
. From Russell and Zermelo to New Foundations 
. “Contradictions Really Scare Me”: New Foundations and the Paradoxes 
. Some Concluding Remarks on the Significance of Consistency Proofs 

PART II QUINE, SET THEORY, AND PHILOSOPHY 

 Quine’s Philosophy of Set Theory 
. Background in Russell 
. Early Developments: From “The Logic of Sequences”
to New Foundations 
. New Foundations as Philosophy of Set Theory 

vii
viii Contents
 Clarifying Our Conceptual Scheme: Set Theory
and the Role of Explication 
. Clarifying Our Conceptual Scheme 
. Explication 
. Quine’s Mature Philosophy of Set Theory: Set Theory and Its Logic 
. An Addendum on Quine and Carnap on Tolerance and Set Theory 

PART III NEW FOUNDATIONS AND THE PHILOSOPHY

OF SET THEORY 
 The Iterative Conception and Set Theory 
. Boolos’s Exposition 
. The Iterative Conception As Set Theory 
. The Iterative Conception and Zermelo’s Axioms 

 New Foundations, the Axiom of Choice, and Arithmetic 

. The Axiom of Choice 
. Arithmetic in New Foundations 
. Exploring the Set-Theoretic Universe 

Bibliography 
Index 
 Preface

This book emerged out of my doctoral dissertation at the University of

Illinois at Chicago, written some years ago under the direction of W. D.
Hart. After completing the dissertation, I moved on to researching topics
of a more general nature in Quine’s philosophy, though always with
an eye toward how his early work in logic and the foundations of mathe-
matics shaped his general approach to philosophy. From time to time,
I continued to present some of my views on Quine’s philosophy of set
theory and its contrasts with much contemporary philosophy of set
theory. I remained surprised by – in discussions of the philosophy of
set theory – how set theory was just about always and without question
identified with Zermelo-Fraenkel set theory and its variants and the asso-
ciated iterative conception of set. From this, it seemed to me that there
might still be something to contribute to contemporary discussions of
the philosophy of set theory by focusing on Quine’s alternative
approach – an approach that ignores the iterative conception, at least as a
metaphysical view about sets, and that is a good deal more experimental
and exploratory in nature, seeing set theory as a still largely unsettled
area of the mathematical sciences.
In assembling the following list of acknowledgments, I could not help
being reminded of the words of a recent Nobel laureate:
Some are mathematicians
Some are carpenters’ wives …
We always did feel the same
We just saw it from a different point of view.
The support and inspiration I drew throughout the writing of this book,
teachers, friends, and colleagues supported and inspired me in ways that
were as varied as they were indispensable. The paths that many of these


Bob Dylan, “Tangled Up in Blue,” Blood on the Tracks (New York, NY: Columbia Records, ).

ix
x Preface
people chose are very different from mine, but without their support this
project never would have been finished. First of all, I would like to thank
my dissertation committee Juliet Floyd, W. D. Hart, Peter Hylton,
Colin Klein, and Daniel Sutherland for their input into the dissertation
that led to this book. W. D. Hart and Daniel Sutherland were especially
helpful in continuing to urge me to rework the dissertation into a book,
and Peter Hylton provided me with very thorough and detailed com-
ments on the original dissertation. For encouragement and comments on
portions of the book manuscript, I thank Gary Ebbs and Warren
Goldfarb. I also thank Timothy Bays, Patricia Blanchette, Michael
Detlefsen, and Curtis Franks for allowing me to present some of this
material at the  Midwest PhilMath Workshop. Dean Joan Foster
provided me with research funds for this project by way of multiple
Metropolitan State University of Denver (MSU) Liberal Arts and
Sciences (LAS) mini-grants, and Rebecca Dobbin of the Dean’s office
helped me to obtain research materials. I also thank Roy Cook, Thomas
Forster, Craig Fox, Matthew Makley, Nick Robertson (–),
Marcus Rossberg, Paul Roth, Joseph Ullian, the staff of the Houghton
Library at Harvard University, and my colleagues and the staff of the
MSU Denver Department of Philosophy. I was also assisted greatly by
comments on an early portion of the manuscript from two anonymous
referees for Cambridge University Press. I also thank Hilary Gaskin at
Cambridge University Press for her interest in this project as well as for
her patience and understanding as I completed my manuscript.
Although I have mentioned him already, I want to single out Bill
Hart, my dissertation advisor and friend. His confidence in and support
of my work have always kept me working. I thank also his wife, Faith,
for her support and encouragement. My friend Kimo Quaintance
deserves a special mention for the many conversations we have shared
over the years and for helping me to see connections where there seemed
to be none. In addition, a late-night conversation with my friend Ben
Nichols was crucial to reigniting my interest in this project.
Finally, I would like to thank my family, including Bill Burke, my
mother, Mary, my father, Paul (–), and my brother, Dan.
They have all, in various ways, helped me to complete this book. Most
of all and most important, I thank my wife, Haewon, and our son, John.
They have made this work possible and have given more to me than I
ever could have asked for. Now that it is done, I hope we will all be
spending a lot more time at the park together.
 Introduction

Since the late s, there has been widespread agreement among both
philosophers and mathematicians that the only viable conception of a set
is that of the iterative conception as embodied in Zermelo–Fraenkel (ZF)
set theory and its related systems. Indeed, in any discussion, set theory
is now treated as practically synonymous with ZF set theory, and the
iterative conception is treated as something near essential to the very nat-
ure of sets. Other approaches to set theory are generally neglected if not
dismissed outright in set-theoretic research. However, this has not always
been the situation, as I hope to show in this book by examining W. V.
Quine’s set theory, New Foundations (NF), and its place in the philoso-
phy of set theory more generally. I should make clear at the outset that
what follows is not meant to be an argument in favor of NF over other
set theories. Rather, the point will be that work in set theory generally,
at least in its current state, should be conducted in the more pluralistic
and pragmatic way that I take to be characteristic of what I will identify
as the approach to set theory as explication. I describe the contrasting
approach, which aims to discover a single correct notion of set, as set
theory as conceptual analysis. Let me begin first, though, with some
very general history that will be further developed in the main body of
the text.


The iterative conception of set had received some attention in the literature on the foundations of
set theory, but its first thorough treatment came in George Boolos, “The Iterative Conception of
Set,” reprinted in his Logic, Logic, Logic, ed. Richard Jeffrey (Cambridge, MA: Harvard University
Press, ), pp. –.

Admittedly, few if any philosophers of set theory talk of capturing the essence of a set; essence has
become a fairly unfashionable notion in contemporary philosophy with its drive toward scientific
methods and naturalism. Still, it seems to me that the near dogmatic adherence to the iterative
conception suggests a commitment to it yielding something like the essence of a set. It is not hard
to find examples of the view that sets must be like this, where the “this” is filled in by the iterative
conception. We will see an example of the sort of view I have in mind in Chapter  in discussing
Boolos’s account of the iterative conception.


 Introduction
Around , in light of the discovery of the set-theoretic paradoxes,
the concept of set had to be rethought. Up to that point, this concept
had been accepted as basically well understood, taken to be what philoso-
phers and logicians had traditionally meant by the extension of a concept
or of a predicate. The realization that not every predicate could deter-
mine an extension, on pain of paradox, left set theory at somewhat of a
loss in accounting for its fundamental notion. The idea that a set is the
extension of a predicate could be captured formally by what has become
known as the comprehension principle, “(∃x)(∀y) (x ∈ y ≡ φ),” where x
is not free in the formula “φ.” But if we let “φ” be “y ∉ y,” for example,
we can then very easily derive Bertrand Russell’s paradox of the set of all
non-self-membered sets. If this set is a member of itself, then it is not a
member of itself; and if it is not a member of itself, then it is a member
of itself – hence a contradiction. This principle would have to be some-
how restricted; but beyond the intuitive thought that a set is the extension
of a predicate, which then also involves a commitment to sets to being
extensional entities, the tradition did not have much more to go on.
Instead of looking for something inherent in the notion of a set that
would yield an appropriately restricted existence principle, logicians and
set theorists focused on what they took to be the mathematical value of
the theory. In particular, set theory could provide a rigorous foundation
for all of ordinary mathematics. The various number systems and opera-
tion on them, the notions of a relation and of a function, and even
geometry could all be reduced to sets along with the sole relation of mem-
bership and logical vocabulary. In addition to this, set theory had to pro-
vide an account of the mathematics of the infinite, since Georg Cantor’s
discovery of an apparently consistent arithmetic of infinite numbers had
been the real advance gained for mathematics by way of set theory.
Previously, mathematicians had mostly thought any sort of actual infinite
to be inherently contradictory. They had operated instead with only
potential infinites. In its most basic form, this is the idea that we could
always add  but that this yielded only another – though larger – finite
number. Cantor’s development of a rule-governed arithmetic of the infi-
nite in a sense legitimized the infinite for the mathematical community.
Hence a successful reworking of set theory was to restrict the comprehen-
sion principle enough to prevent the paradoxes but not so much that

This is the simplest of the set-theoretic paradoxes, and it is perhaps for this reason that is the best
known among them. There are other paradoxes, of course, which I will discuss further in the text.
In particular, there are also Cantor’s paradox of the largest cardinal number and the Burali-Forti
paradox of the largest ordinal number.
 Introduction 
it would lack the power it needed to capture ordinary mathematics as
well as the mathematics of the infinite. Such broad criteria left much
freedom for choosing how to do this. As we will see, the two main
approaches that emerged were Ernst Zermelo’s set theory and its related
systems and Russell’s theory of types. Both were considered to be
legitimate approaches to set theory, and both were used widely in foun-
dational studies for mathematics through at least the early s; after
that, Zermelo’s theory rose to dominance. Neither was considered to
capture anything like the essence of a set unless the essence is minimally
understood as including some restricted version of the comprehension
principle, an axiom of extensionality (sets are identical when they have
the same members; this is a distinguishing feature of sets, so much so
that it would be hard to consider a theory that lacked it as set theory),
and enough strength to capture what made set theory mathematically
interesting in the first place.
This period of set theory up through the s, then, was largely a
pragmatic undertaking, involving much exploration and experimentation
and a certain open-mindedness about how set theory should be done. In
addition to the main systems of Russell and Zermelo, many variants of
these systems were also studied. There was much interest in whether
these different set theories could decide certain fundamental questions
about sets that were still open, such as the axiom of choice (AC) or
Cantor’s continuum hypothesis. There was also a general interest in the
comparative study of the various options: How strong were they in com-
parison to each other, how much mathematics could they prove given
their various assumptions, or how much could they tell us about the infi-
nite? Aside from these main systems, there were other sets theories that
diverged from them, though they, too, would adhere to the basic criteria

It was dominant at least as a set theory. Russell’s type theory tended to get relegated to logic as a
higher-order logic. Although this is an important and interesting topic, I will not consider it further
here, as it would take me well beyond my primary concern with the philosophy of set theory.

Many of the standard texts of this period clearly reflect this approach to set theory. See, for exam-
ple, the many set theories studied in A. A. Fraenkel, Y. Bar-Hillel, and A. Levy’s Foundations of Set
Theory, nd rev. edn (Amsterdam: North-Holland, ), or Mendelson’s Introduction to
Mathematical Logic (Boca Raton, FL: CRC Press, ). As we will see, Quine’s Set Theory and Its
Logic, rev. edn (Cambridge, MA: Harvard University Press, ), is certainly part of this tradition
in set theory. This is quite different from most contemporary set theory texts, which focus pretty
strictly on Zermelo’s set theory and its variations. This may now be starting to change again in
light of the growing interest and value found in the study of non-well-founded set theories. Some
standard introductory set theory texts are starting to cover this topic; see, for example, Keith
Devlin, The Joy of Sets: Fundamentals of Contemporary Set Theory, nd edn (New York, NY:
Springer-Verlag, ), Ch. , and Karel Hrbacek and Thomas Jech, Introduction to Set Theory,
rd rev. and expanded edn (New York, NY: Marcel Dekker, ), Ch. , Sec. .
 Introduction
that had been set for a viable set theory. Such is the set theory that is the
main topic of this book: Quine’s New Foundations.
Quine first put forward his NF in the  paper for which this set
theory is named. It is strikingly different from most other set theories in
that it allows for a universal set, that is, a set of all sets. Such “big” sets
as this one, along with the set of ordinal numbers and the set of cardinal
numbers, have often been thought to be the source of the set-theoretic
paradoxes of Russell, Cesare Burali-Forti, and Cantor, respectively, and
so have generally not received widespread attention in the literature. But
as we will see in the opening chapters, such sets were not always so easily
dismissed or altogether ignored. Early practitioners did consider such
sets – not always favorably, but they were certainly not far from main-
stream thinking about set theory, whereas they typically are thus
positioned these days. Questions about these big sets were a real concern,
raising questions that needed to be answered if set theory was to go on.
For example, if big sets were the source of the paradoxes, on what
grounds could they be ruled out as illegitimate? Or if they were to be
included as part of the set-theoretic universe, how could they exist with-
out giving rise to contradiction? These were genuine and important
issues to consider during set theory’s founding and development by
Cantor, Russell, and Zermelo. In the more recent development of set
theory, these issues were not so much resolved as just simply faded from
view as the two main set theories – ZF set theory and Russell’s theory of
types – emerged, both denying the existence of these big sets. With
this, we lost any mathematical investigation into what Cantor identified
as the absolute infinite. As I have already noted, this early period of set
theory was not one of dogmatism, and research in set theory remained
fairly pluralistic through the s. With many important questions still
open, chief among them being the continuum hypothesis, no one axio-
matization of set theory was taken as definitive, and much research


W. V. Quine, “New Foundations for Mathematical Logic,” American Mathematical Monthly :
(), pp. –.

As we will see later, it is not quite as deviant as it is often portrayed to be. While it certainly does
behave in unusual ways, particularly when its “big” sets are under consideration, it also develops
fairly naturally out of insights gathered from Russell and Zermelo. We will see that, in a way, it is
a generalization of ideas that Quine took from both of them.

In his early work, Russell himself thought it near obvious that these big sets must exist, and as we
will see, it was this belief that brought him to the discovery of Cantor’s paradox.

Cantor himself did not think that the absolute infinite could be investigated mathematically, but
for theological rather than mathematical reasons.
 Introduction 
concerned comparative accounts of the various axiomatizations that were
available. It was in this context of pluralism and tolerance that Quine
introduced NF.
This book consists of seven chapters, divided into three main parts,
though all related in some way. The first part of the book is concerned
primarily with the early development of set theory, including the emer-
gence of Quine’s NF. In Chapter , I present a brief history of set theory
from its beginnings with Cantor through to the discovery of the para-
doxes, and then, in Chapter , I turn to the main proposals offered by
Zermelo and Russell for resolving them. In these first two chapters,
I aim to bring out two important points that I mentioned previously. The
first is that to the extent to which there was any single shared conception
of a set at the founding of the theory, it was the idea that a set is the
extension of a predicate. The second point is that after the paradoxes,
the success of a set theory was to be judged by its ability to explain the
infinite and to serve as a framework for reconstructing ordinary non-set-
theoretic mathematics. This gives rise to the approach to the philosophy
of set theory that I describe as set theory as explication. This is the idea
that there is no uniquely correct notion of a set beyond its being the sort
of object that fulfills the minimal criteria that I have mentioned already:
It is an extensional entity, and it must somehow be restricted enough to
avoid the paradoxes but not so much so that it ceases to be capable of
fulfilling its intended role in mathematics. I take this notion of an expli-
cation from Quine, who himself took it, as we will see, from Rudolf
Carnap and Russell. Quine himself states in his book Word and Object
that for an explication, “[w]e fix on the particular functions of the
unclear expression that make it worth troubling about, and then devise a
substitute, clear and couched in terms to our liking, that fills those func-
tions.” In this view, I argue, set theory is developed pragmatically in an
experimental and exploratory spirit. This approach is to be contrasted
with what I describe as set theory as conceptual analysis. This view takes
it as a given that there is a single correct set theory and, through concep-
tually analyzing the notion of set itself, we can discover what it is. This
latter view appears to be dominant in the contemporary philosophy of
set theory, with its adherence to the iterative conception of set and the
general unwillingness to entertain set theories that run contrary to this


Fraenkel and Bar-Hillel’s Foundations of Set Theory, first published in  with its last edition
published in , is a classic of this sort of work in set theory.

W. V. Quine, Word and Object (Cambridge, MA: MIT Press, ), pp. –.
 Introduction
conception. We will see examples of this attitude mostly in later
chapters, where I discuss NF and its accompanying philosophy in the
context of the iterative conception.
In Chapter , which concludes the opening line of thought with its
historical focus on the paradoxes, I present Quine’s NF as combining the
insights of both Zermelo and Russell into resolving the paradoxes. Here,
I emphasize that Quine’s exploratory and experimental approach fits
squarely within the tradition of explication. Thus NF is far from an
anomaly in the history of set theory, though it is often treated as such
within the context of contemporary philosophy of set theory. In a way,
then, the current dominance of the iterative conception is much more
the anomaly in the history of set theory.
The second part of the book comprises two chapters in which I focus
more directly on Quine’s philosophy of set theory within the context of
his philosophy as a whole. The focus here is not so much on NF specifi-
cally as on the philosophical thinking that informed Quine’s approach to
set theory generally. In Chapter , I focus primarily on how Quine’s
philosophy of set theory developed out his early engagement with
Russell’s logicism, particularly as put forward in Russell’s Principles of
Mathematics and the more technical working out of this project in
Whitehead and Russell’s Principia Mathematica. Whereas Chapter 
emphasizes Russell’s foundational work as part of the tradition of set
theory as explication, Chapter  tells a broader story, emphasizing some
of the tensions between Russell’s mathematical and philosophical
approaches to the foundations of mathematics. It is in the former where
we find the idea of set theory as an explication. I argue that we can
understand Quine’s own particular conception of set theory as explica-
tion as emerging out his attempt to come to terms with the tensions in
Russell’s work. The final part of Chapter  returns to NF to emphasize
the philosophy that accompanies it. In particular, we see in Quine an


There is nothing in the iterative conception that demands that it be approached from the side of
conceptual analysis. After all, as I have already remarked, the iterative conception is now most
commonly taken to be embodied in the axioms of Zermelo’s theory, which originally emerged
out of the explicative approach. Indeed, I take it that there are good mathematical reasons to now
favor a set theory such as ZF, but these are not the reasons that are typically offered in favor of
the iterative conception over other conceptions of set. More typical are appeals to its rather intui-
tive character. This is an argument that I reject, as we will see throughout this book in various
ways.

Bertrand Russell, Principles of Mathematics, nd edn (London: George Allen and Unwin Ltd.,
[first edition ]). Alfred North Whitehead and Bertrand Russell, Principia Mathematica.
 vols. nd edn (Cambridge: Cambridge University Press, ,  [first edition , ,
]).
 Introduction 
attitude that is very critical of what he identifies as the more metaphysical
approaches to the philosophy of set theory.
In Chapter , I continue to focus on the philosophy that informs
Quine’s approach to set theory, but I turn to his more developed and
more general approach to philosophy as a whole. The main theme here
will be how Quine takes the simplification and clarification of our con-
ceptual scheme as a main task for philosophy. His work in set theory is
largely a contribution to this project, focusing on the mathematical por-
tion of our conceptual scheme and reaching its culmination in Set Theory
and Its Logic. Within the context of Chapter , I also discuss in some
detail Quine’s understanding of the notion of explication. Since my
account of Quine often pushes him very close to Carnap, I conclude
Chapter  by trying to locate where their differences lie, particularly with
regard to their appeals to tolerance in their respective philosophies.
In Part III, I consider in more detail the other tradition in the philoso-
phy of set theory, that of conceptual analysis, by examining in Chapter 
some arguments in favor of the iterative conception of set, most notably
championed by George Boolos. I argue that this preference ultimately
rests on pragmatic considerations that support NF equally well. I then
look at NF in more detail in this context to show that, like ZF-style set
theories, it largely satisfies the criteria offered earlier for judging a set the-
ory successful. That is, it appears to eliminate the paradoxes while also
providing an account of the infinite and serving as a framework for
reconstructing the rest of mathematics within it. In addition, I emphasize
that contemporary set-theoretic research tends to focus on set theory
itself as a subject for mathematical investigation and thus no longer limits
itself to being merely a foundation for other branches of mathematics.
NF, with its “big sets” such as the universal set, the set of cardinal num-
bers, and the set of ordinal numbers, is thus more consonant with
current mathematical practice regarding set theory than might have been
thought. It gives us a way to investigate the mathematical behavior of
such sets, which ZF and related systems exclude. Again, my ultimate aim
is not to argue for NF as the correct set theory over other set theories
but to show that the concurrent development of various set theories
furthers our understanding of the set-theoretic universe and so also our
mathematical knowledge generally.
In Chapter , I examine some potential problems with NF. NF’s
development of arithmetic does not appear to proceed as elegantly as
that of ZF. One way to deflect this challenge is to draw attention again
to the fact that set theory has moved far beyond being merely a
 Introduction
foundation for other branches of mathematics. I then also observe that
one reason why ZF does give us an elegant account of, say, arithmetic is
that ZF begins by ruling out certain troublesome sets from the start.
In the more pragmatic view that I am putting forth, this a perfectly
acceptable starting point. But I then argue that it would also be perfectly
acceptable not to rule out these sets, particularly when our interest is in
the mathematics of the set-theoretic universe as a whole. Within this
context, I also consider the status of AC in NF. NF is notorious for dis-
proving this axiom, which is used crucially throughout mathematics.
However, I show that – much as in the case of arithmetic – this apparent
failing arises only because NF includes sets that ZF does not. In NF, AC
fails only in full generality. Appropriately restricted, enough of AC is pre-
sent to satisfy the usual mathematical needs. Again, I show that such a
failing of AC is hardly out of line with the set-theoretic tradition. For
much of its history, AC has been treated carefully by set theorists, often
flagging where it is used in proofs or in trying to re-prove results so that
they do not rely on the axiom. Although much of contemporary set
theory has come to treat AC as constitutive of the behavior of sets, even
considering the principle obvious, this seems unfounded. After all, AC is
well known for its highly unintuitive consequences, such as the Banach–
Tarski paradox. In consequence, the truth of AC seems far from settled,
thus leaving as a worthwhile endeavor the investigation of set theories for
which it fails.
My aims in the book are broadly philosophical in that, most generally,
I want to urge that logic and mathematics continue to be fruitful sources
for thinking about philosophical problems that have been with us since
philosophy’s beginning. In particular, this book raises again the age-old
question of how we come to acquire our nonempirical knowledge. In
arguing that set theory is pragmatically developed, I want to suggest that
mathematics in its foundations develops in ways that share much with
the natural sciences and therefore is not necessarily the paradigm of a
priori knowledge that philosophers have often thought it to be.
Before concluding this introduction, I should comment further on
what I have been describing as an approach to set theory as explica-
tion. I have emphasized that this approach to set theory is pragmatic.
Some of my exposition up to this point and in what follows may sug-
gest that the pragmatic approach to set theory is unique to my account
of set theory as explication and that it no longer has much hold on
contemporary philosophy of set theory. This is not entirely true. A
number of commentators have noted that this tradition has a firm
 Introduction 
place in the history of set theory. Furthermore, a pragmatic
approach to set theory is not entirely absent from contemporary set
theory. But in accord with the account that I have been laying out,
this pragmatism happens only within ZF-style set theories. It does not
extend to alternative axiomatizations. This is just the sort of dogma-
tism in set theory that I will question in what follows. With this in
mind, let me now present Quine’s NF and its place in the philosophy
of set theory.


For some examples of the pragmatic tradition in the history of set theory, see Gregory H. Moore,
“The Origins of Zermelo’s Axiomatization of Set Theory,” Journal of Philosophical Logic :
(), pp. –; M. D. Potter, “Iterative Set Theory,” The Philosophical Quarterly :
(), p. ; and Heinz-Dieter Ebbinhaus, Ernst Zermelo: An Approach to His Life and Work
(New York, NY: Springer, ), p. . I should also add that I have previously described what I
am now calling set theory as explication as the pragmatic approach to set theory. However, this
label seemed to cause too much confusion with American pragmatism as a general philosophical
position. For suggesting the new terminology, distinguishing between set theory as explication
and set theory as conceptual analysis, I thank Gary Ebbs.

We find this in Penelope Maddy’s recent Defending the Axioms: On the Philosophical Foundations
of Set Theory (New York, NY: Oxford University Press, ), in her discussions of intrinsic ver-
sus extrinsic justifications of the axioms of set theory. The former lines up with conceptual analy-
sis and the latter with explication.
 PART I

Set Theory’s Beginnings

 CHAPTER 

Cantor and the Early Development of Set Theory

This opening chapter provides a basic account of the development of set

theory up to about  and the discovery of the set-theoretic paradoxes.
Here, I focus specifically on the most important results of Georg Cantor
(–), the founder of set theory. Two important themes begin to
emerge here that will receive greater attention in later chapters. First, we
will see that Cantor developed set theory with the aim of making mathe-
matical sense of the infinite. This comes in two intertwined parts:
Cantor’s attempts to legitimize the actual, or proper, infinite and his dis-
covery of countable and uncountable infinities. Of particular importance
is Cantor’s effort to legitimize the infinite as actually existing by provid-
ing mathematical rather than philosophical arguments. This will be espe-
cially important for my discussion in Chapter  of how the notion of set
was eventually constrained to consistency after the paradoxes. The second
theme, which is most important for my overall aims, is that Cantor
developed set theory as a mathematical theory, addressing concerns
within mathematics and not about some prior philosophical conception
of a set.
This will continue to be a major theme throughout this book and one
that I will develop most directly in Chapter  in considering W. V.
Quine’s particular view of the philosophy of set theory. Briefly put for
now, I describe this tradition in the philosophy of set theory as set theory
as explication. This will be contrasted with what I will later describe as
set theory as conceptual analysis. This latter tradition is now much more
prominent in the philosophy of set theory and can be roughly character-
ized by the view that there is a uniquely correct conception of set theory
and we will be able to discover it by way of conceptual analysis of the
notion of set. Contemporary philosophers of set theory have largely
followed this path and are now in general agreement that the iterative
conception as embodied in the axioms of Zermelo–Fraenkel set theory
is what is meant by set theory. The other tradition – set theory as

 Quine, New Foundations, and the Philosophy of Set Theory
explication – has been largely forgotten, if it has ever really been
accorded its place as a distinct tradition at all. Quine is perhaps the most
prominent adherent of this view. Rather than aiming at a single correct
theory of set, it attempts to identify what made set theory worthwhile in
the first place and develops a set theory that is in accord with these fea-
tures. In doing so, it allows for a plurality of set theories and is explora-
tory and experimental in nature. It does not, however, rule out our
eventually settling on a single set theory. However, doing so would be
largely the result of certain pragmatic concerns and not because we had
somehow arrived at, say, the essence of sethood. We will have to wait
until Chapter  and beyond to fully develop these two traditions and
examine their relative philosophical merits. Let us instead consider set
theory at its very beginnings.

. The Beginnings of Set Theory as a Mathematical Discipline

Set theory began as a mathematical discipline – though often deeply con-
nected to the philosophical views of its founders – during the nineteenth
century as the calculus was being treated to greater rigorization. During
this period, troubles began to arise around the unrestricted use of func-
tions and the related notions of convergence and continuity. Since the
beginnings of the calculus with Gottfried Leibniz and Isaac Newton, the
notion of a function had been gradually expanded from analytic expres-
sions to arbitrary correspondences. By the late eighteenth century, how-
ever, Leonhard Euler, along with Joseph-Louis Lagrange, had made the
first significant expansions of the concept of a function by introducing
methods relating to infinite series. Before their work, mathematicians
had held to the view that a function must always have the same analytic
expression regardless of its domain. The expansion of this concept
allowed for a function to be expressed differently depending upon its
domain. As the use and variety of what was to count as a function con-
tinued, mathematicians gradually adopted a broader understanding of
the concept of a function itself. But with this came new difficulties
regarding the properties of functions, chief among them what was meant
by continuity, differentiability, and integrability. Furthermore, many of
these concepts had no more than an intuitive foundation expressed in

Akihiro Kanamori, “The Mathematical Development of Set Theory from Cantor to Cohen,” The
Bulletin of Symbolic Logic : (March ), pp. –. I follow Morris Kline, Mathematical Thought
from Ancient to Modern Times (New York, NY: Oxford University Press, ), pp. – for
the account of the rigorization of the calculus that follows over the next few pages.
 Cantor and the Early Development of Set Theory 
temporal, dynamical, or geometric terms. Mathematicians of what has
become known as “the critical movement” began urging that a founda-
tion for analysis be constructed more rigorously and from arithmetical
concepts. While geometry had frequently been used in the past for such
foundational work, the rise of non-Euclidean geometries in the nine-
teenth century brought into question the privileged status that geometry
seemed to hold. If any branch of mathematics remained primary, it was
generally thought to be arithmetic. Central among those urging analysis
in this direction were Bernard Bolzano, Augustin-Louis Cauchy, Gustav
Dirichlet, and Karl Weierstrass.
By the second half of the eighteenth century, Euler and Lagrange had
taken a function to be continuous when the same expression held and
discontinuous at points where the expression changed. However, this
was not the modern understanding of these notions or of a function gen-
erally. Dirichlet gave the modern definition of a function in , stating
that y is a function of x when, for each value of x in a given interval,
there is exactly one value of y. In line with mathematicians’ acceptance
of an increasingly broad conception of a function, he added that within
the given interval y could depend upon x according to more than one
law and that the dependence did not have to be expressed by way of
mathematical operations. With the concept of a function so refined, a
proper understanding of continuity and discontinuity also emerged,
primarily as the result of work by Bolzano and Cauchy. Bolzano defined
a function f (x) as continuous in an interval if at any x in the interval,
the difference f (x + ω) – f (x) can be made as small as one wishes by
taking ω sufficiently small. Notable in his definition is the elimination of
infinitesimals – infinitely small but nonzero quantities – used throughout
the eighteenth century but with no rigorous basis. Cauchy similarly
defined the notion of continuity while denying infinitesimals. A key
innovation here was his introducing the notion of a limit. He replaced
talk of infinitesimals with talk of limits describing a variable quantity as
becoming infinitely small when its numerical value decreases indefinitely
so as to converge to a limit zero. He identified these variable quantities
as infinitesimals but without the metaphysical overtones associated with
their use in Leibniz and his followers. Weierstrass contributed further to
rigorizing the calculus by removing talk of a variable approaching a limit.


Infinitesimals returned with a rigorous basis in the twentieth century in the work of Abraham
Robinson, first published in his “Non-Standard Analysis,” reprinted in his Selected Papers, vol. ,
eds. H. J. Keisler et al. (New Haven, CT: Yale University Press, ), pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
He stressed instead that the variable was just a letter standing for any
one of a set of values that may be assigned to the variable. In this way,
he eliminated temporal and dynamic intuitions from the rigorous formu-
lation of the calculus.
The derivative, too, stood in need of a clear definition. Mathematicians
before the nineteenth century appealed to geometric intuitions and the
mysterious notion of an infinitesimal. Again, Bolzano gave the first clear
account in , defining the derivative of f (x) as the quantity f (x), which
the ratio [ f (x + Δx) − f (x)]/Δx approaches indefinitely close as Δx
approaches zero through positive and negative values. Cauchy added to
this definition by unifying it with Leibnizian differentials, defining dx as
any finite quantity and dy as the derivative f (x)dx so that the differentials
are accounted for in terms of derivatives. Finally, he clarified the relation-
ship between Δx/Δy and f (x) by way of the mean value theorem: Δy =
f (x + θΔx), where  < θ < . While the work of Bolzano and Cauchy
yielded rigorous accounts of continuity and the derivative, this still left
questions about continuous functions and differentiability. In particular,
it had previously been thought that all continuous functions must be dif-
ferentiable, but in , Bolzano found an example of a continuous
function with no finite derivative at any point, though he did not pub-
lish it. However, Bernhard Riemann did publish an example of such a
function in , forcing the mathematical community to recognize that
continuity and differentiability had to be distinguished. Such counterex-
amples to previous thinking on these topics led mathematicians to be all
the more skeptical of appeals to intuition and geometrical approaches to
the calculus.
These developments led also to concerns over the integral. Newton
had shown how the integral could be found by simply reversing differen-
tiation, which is still the standard approach. But now questions arose
about the integral for discontinuous functions. Leibniz had already exam-
ined calculating areas and volumes in terms of a sum of elements such as
rectangles and cylinders. Eighteenth-century mathematicians had fol-
lowed him in this, but again with the usual lack of rigor. Cauchy began
to correct this. Joseph Fourier had already shown that the Leibnizian
view worked to calculate the integral for discontinuous functions, but
this still left the problem of expressing the integral analytically for discon-
tinuous functions. Cauchy took up this problem, emphasizing that that
integral should be given as the limit of a sum rather than in the
Newtonian way as the inverse of differentiation. His aim, achieved in
, was to show that the integral could be given arithmetic expression.
 Cantor and the Early Development of Set Theory 
Ultimately, however, his methods proved too restrictive to apply to the
many irregular functions that had entered into analysis with the broaden-
ing of the concept of a function. Riemann stepped in to correct this in
an  paper on the trigonometric series in which he sufficiently gener-
alized the notion of an integral, using an example of an integrable func-
tion with an infinite number of discontinuities in every arbitrarily small
interval to show the success of his definition. Some outstanding issues
remained, but by , the integral had received sufficiently rigorous
and general expression.
We have seen here how Bolzano, Cauchy, Dirichlet, and Weierstrass,
among others, addressed the various foundational problems in the calcu-
lus by further articulating such concepts as convergence and continuity
and by replacing infinitesimals with the notion of a limit in terms of the
now familiar epsilon-delta language. The result of their work not only
eliminated the apparent flaws in the calculus but, more important for
our purposes, also restored an emphasis on foundational work in mathe-
matics and on deductive rigor in particular. Making sense of these new
functions in terms of an infinite series was achieved only through their
careful specification by means of deductive methods. Cantor emerged
from this tradition that restored proof as the focal point of mathematics,
thus furthering the greater abstraction and generality characteristic of
modern foundational work in mathematics. Indeed, Cantor also made
direct contributions to the rigorization of the calculus, and his founding
of set theory emerged directly out of this research. Perhaps his most funda-
mental contribution was an account of the real numbers themselves. For
most of the developments toward rigorization that we have seen so far, the
structure of the real number line was assumed to be well enough under-
stood. But as we have seen, greater clarification of one concept often
exposed a lack of clarity in another. This was also the case for the real num-
bers themselves. For example, Cauchy was unable to prove his criterion for
the convergence of sequence sufficient specifically because the structure of
the real numbers themselves had not yet been subjected to rigorization. It
was thought that an intuitive understanding was enough. We will return
to this topic soon enough, but let us begin with Cantor’s earliest mathema-
tical contributions to the emergence of set theory.
Cantor’s first important move toward set theory came in  in
proving a uniqueness theorem for trigonometric series: If a trigonometric


Kanamori, “The Mathematical Development of Set Theory from Cantor to Cohen,” p. .

Kline, Mathematical Thought, p. .
 Quine, New Foundations, and the Philosophy of Set Theory
series converges everywhere to zero, then all of its coefficients are equal
to zero. He generalized this result in his “On the Extension of a
Theorem of the Theory of the Trigonometric Series” [] to obtain
the result: For a collection of real numbers P, let P′ be the collection of
limit points for P, and let P(n) be the result iterating this operation n
times. If a trigonometric series converges everywhere to zero except on a
P where for some n, P(n) is empty, then all of its coefficients are equal to
zero. While mathematically significant in itself, Cantor’s result had
important consequences for the beginnings of set theory. His key move
here was in considering collections of real numbers specified by an opera-
tion. Here, Cantor moves beyond considering just individual limit points
to considering collections of them. In specifying such collections by an
operation (later, by a law), Cantor was not far from the idea that sets are
the extensions of concepts, which located the notion of a set as the exten-
sion of a concept deep within the logical tradition. This was, of course,
as we will see, the notion of a set that was to yield the paradoxes but was
to be maintained in the post-paradox world in a sufficiently restricted
sense – restricted enough to restore consistency yet preserve all that was
mathematically valuable in the theory.
Just a couple of years later, Cantor made another important move
toward the full development of modern set theory: He showed how to
build up the real numbers from collections of rational numbers. While
Richard Dedekind’s cuts are perhaps now the more familiar construction
of the real numbers from rational numbers, Cantor construed the real
numbers as what he called “fundamental sequences” of rational numbers,
or what are now often called Cauchy sequences. Elaborating on the
motivations for this work some years later in his  Grundlagen,


Georg Cantor, “Über einen die trigonometrischen Reihen betreffen Lehrsatz,” in Cantor,
Gesammelte Abhandlungen: Mathematische und Philosophischen Inhalts, ed. Ernst Zermelo
(Hildesheim: Georg Olms Verlagsbuchhandlung, ), pp. –.

Cantor, Gesammelte Abhandlungen, pp. –.

Kanamori, “The Mathematical Development of Set Theory from Cantor to Cohen,” p. . On the
relationship between sets as the extensions of concepts and the specification of a set by an opera-
tions, see José Ferreirós, Labyrinth of Thought: A History of Set Theory and Its Role in Modern
Mathematics (Boston, MA: Birkhäuser Verlag, ), pp. –, . See also George Boolos,
“The Iterative Conception of Set,” in Logic, Logic, and Logic, ed. Richard Jeffrey (Cambridge, MA:
Harvard University Press, ), pp. –. I will return to this point in more detail in later
chapters.

Cantor, Gesammelte Abhandlungen, pp. –.

The fundamental sequences are defined by the same property Cauchy used for his criterion of con-
vergence; see Ferreirós, Labyrinth of Thought, p. .
 Cantor and the Early Development of Set Theory 
Cantor singles out three definitions of the real numbers: Weierstrass’s,
Dedekind’s, and his own. Remarking on Weierstrass’s definition, Cantor
observes:
One sees here that the creative element which binds the set with the num-
ber defined through it lies in the formation of sums; but it must be empha-
sized as essential that only the summation of an always finite Anzahl of
rational elements is used and that the number b to be defined is not set at
the beginning as equal to the sum Σaν of the infinite series (aν); this
would be a logical error, because the definition of the sum Σaν is only
reached by equating it with the finished number b which is necessarily
defined earlier. I believe that this logical error, which was first avoided by
Weierstrass, was in earlier times the universal practice, and was not
noticed because it belongs to those rare cases in which actual errors can
do no significant harm to calculations.—Nevertheless I am convinced that
all the difficulties which have been found in the concept of the irrational
are connected with the indicated error, whereas if we avoid this error the
irrational numbers take root in our minds with the same precision, dis-
tinctness, and clarity as the rational numbers [Cantor’s italics].
Of particular interest here is the “logical error” that Cantor describes,
which will be a theme prominent in the development of set theory, as
we will see in following chapters. The basic idea in Cantor’s logically cor-
rect approach is that we do not first assume the existence of a rational
number and then try to find some set of objects that will serve as its defi-
nition. Rather, we assume only the antecedently accepted objects – in
this case, the rational numbers – and define the new objects, the real
numbers, in terms of them. There is not, then, a preexisting set of
objects, the real numbers, of which we are trying to get the single correct
account. Instead, we identify the key mathematical properties of the real
numbers – those properties that make the real numbers worthwhile –
and then try to construct them out of antecedently well-understood
objects. Indeed, Cantor goes on to show that all three of the proposed
definitions do have the mathematical properties of the real numbers. By
doing so, we bestow the real numbers “with the same precision, distinct-
ness, and clarity as the rational numbers.” Furthermore, the three defini-
tions are to be distinguished only in terms of their utility (and Cantor
preferred his own definition on these grounds); otherwise, they all have


Cantor, Foundations, sec. , para. . I have used Ewald’s translation and so have left “Anzahl”
untranslated. While there is some dispute over how to best translate this term, it can be basically
understood as “ordinal number.”

As we will see, this is very much the attitude that Quine adopts toward the notion of explication.
 Quine, New Foundations, and the Philosophy of Set Theory
equal right to the claim of being the real numbers. Kanamori notes
three particularly important aspects of Cantor’s construction of the real
numbers for his further developing of set theory. The construction in
terms of fundamental sequences led him to focus more explicitly on infi-
nite collections as such, to view them as unitary objects, and finally, to
allow for arbitrary possibilities of such objects, which connects back to
Cantor’s idea of specifying sets by an operation or rule.
We should note two further things about Cantor’s defining of the real
numbers that are particularly important to my own aims. One is his
emphasis on construction so as to avoid the “logical error” of previous
definitions. This approach to the foundations of mathematics becomes
especially prominent in Bertrand Russell’s work, with his emphasis on
logical construction. The other, which is even more important for my
overall aims, arises in Cantor’s comparison of the three different defini-
tions. As long as they all yield the mathematical properties required of
the real numbers, there is no distinguishing among them except perhaps
on pragmatic grounds, such as ease of use, elegance, or simplicity. As we
will see in Chapter , in light of the paradoxes, something like this atti-
tude becomes standard as an approach to set theory itself.
It was Cantor’s next move – to prove that the real numbers are
uncountable – that led to his full-blown development of transfinite set
theory. As Kanamori vividly makes the point, “[s]et theory was born on
that December  day when Cantor established that the collection
of real numbers is uncountable, and in the next decades the subject was
to blossom through the prodigious progress made by him in the theory
of ordinal and cardinal numbers.” During this period, Cantor began
considering infinite iterations of his operator P′, where
′
P ð∞Þ ¼ ∩n ∞ P ðnÞ ; P ð∞þ1Þ ¼ P ð∞Þ ; P ð∞ ⋅ 2Þ ; …; P ð∞ Þ ; …;
2

∞ ∞∞
P ð∞ Þ ; …; P ð∞ Þ
;…
He also began to investigate infinite collections and real numbers and
infinite enumerations as such. Combined, these moves led him to the
basic concepts used in the study of the continuum and to the


Cantor concludes his discussion with just such a comparison of the three definitions and explains
why he prefers his own (paras. –). On this point, see also Joseph Warren Dauben, Georg
Cantor: His Mathematics and Philosophy of the Infinite (Princeton, NJ: Princeton University Press,
), p. .

Kanamori, “The Mathematical Development of Set Theory from Cantor to Cohen,” p. .

Ibid.
 Cantor and the Early Development of Set Theory 
formulation of the transfinite numbers. His first major result in this
direction was to prove that the set of real numbers is uncountable. The
proof first appeared in print in Cantor’s  article “On a Property of
the Totality of All Real Algebraic Numbers.” He first establishes that
the algebraic numbers are countable, where a real number ω is algebraic
if there exists a positive integer n, and integers a, a, …, an, an ≠ ,
such that
an ωn þ an−1 ωn−1 þ ⋯ þ a1 ω þ a0 ¼ 0:
He then proceeds by reductio to show that for any countable sequence
of real numbers, every interval contains a real number that is not in the
sequence:
Let the set of real numbers be countable. Then each real number ω
can be sequenced by indexing them with natural numbers n:
ω1 ; ω2 ; ω3 ; …; ωn ; …
Now, given an interval (α, β) that is a subset of real numbers, it is possi-
ble to find at least one real number η such that η fails to be listed as an
element of the sequence. To find such a number η, let α < β and pick
the first two numbers α′, β′ of the preceding sequence in the interval
(α, β). These form another interval (α′, β′). Then continue this proce-
dure to yield a sequence of nested intervals through (αn, βn), where αn,
βn are the first two numbers of the sequence in the interval (αn− … βn−).
There are two possibilities to consider. If the number of constructed inter-
vals is finite, then at most only one additional element of the sequence could
lie in the interval (αn, βn). By choosing any real number η in the interval
(αn, βn) not equal to the possible number from the sequence, we find a real
number that is not listed in the sequence. On the other hand, if the number
of constructed intervals is not finite, since the sequence α, α′, …, αn, … is
bounded in (α, β), it has an upper limit α∞, and similarly, the sequence
β, β′, …, βn, … has a lower limit β∞. There are two further cases to con-
sider. If α∞ < β∞, then as in the finite case, any real number η in the
interval (α∞, β∞) would be the real number not listed in the preceding
sequence. However, if α∞ = β∞, then η = α∞ = β∞, and η could not be
listed as an element of the preceding sequence. Let η = ωρ. For n a


Ibid.

Georg Cantor, “On a Property of the Totality of All Real Algebraic Numbers,” in From Kant to
Hilbert: A Source Book in the Foundations of Mathematics, vol. II, ed. William Ewald (Oxford:
Clarendon Press, ), pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
sufficiently large index, ωρ would be excluded from all the intervals nested
within (αn, βn). But by virtue of the construction, η must lie within
(αn, βn). This is a contradiction, and hence the collection of real numbers
is uncountable. With this proof, Cantor secured the importance of the
infinite as a legitimate area of mathematical research.

. From the Potential to the Actual Infinite

In his next publication, “A Contribution to the Theory of Manifolds”
[], Cantor began to focus more directly on bijective mappings
between sets. He defined two sets as having the same power if and
only if there is a bijective mapping between them. Whereas his earlier
result on the uncountability of the real numbers showed where such a
mapping failed to hold, he now looked to see where these mappings did
hold. It was these investigations that led him to fully develop the mathe-
matics of the transfinite. At the end of his investigations, however, he
had been able to find only two powers of infinite sets and conjectured
that every set of real numbers has the first power, that is, is countable, or
has the power of the continuum. Cantor’s attempts to solve this early
version of the continuum hypothesis pushed his work forward, leading
him to a more arithmetical approach in studying sets of real numbers as
well to questions of set existence as such. At least where Cantor is con-
cerned, Kanamori observes that “[s]et theory had its beginnings not as
some abstract foundation for mathematics but rather as a setting for the
articulation and solution of the Continuum Problem: to determine
whether there are more than two powers embedded in the continuum.”
We will see this theme continue into Chapter  in the attempts to come
to terms with the paradoxes. We will see there that for Cantor, Russell,
and Ernst Zermelo, what was important in regaining consistency was to
maintain all that was mathematically valuable to the theory. Beyond this,
there was no prior notion of set to be maintained.
Cantor’s next major work – one of the most important in presenting
his mature view of set theory – was “Foundations of a General Theory of
Manifolds: A Mathematico-Philosophical Investigation into the Theory


I have followed Dauben. Georg Cantor, pp. –, in presenting Cantor’s proof, but Kanamori,
“The Mathematical Development of Set Theory from Cantor to Cohen,” p. , also sketches it,
although in less detail.

Cantor, Gesammelte Abhandlungen, pp. –.

Kanamori, “The Mathematical Development of Set Theory from Cantor to Cohen,” pp. –.

Ibid., p. .
 Cantor and the Early Development of Set Theory 
of the Infinite” [], the Grundlagen. In this work, he introduces his
theory of ordinal numbers and the notion of well-ordering. He shifts
focus away from the infinitely indexed operator P′ used in his paper on
the trigonometric series and turns his attention to the indexes themselves,
that is, to what become his ordinal numbers. He also makes a notable nota-
tional change in moving from the symbol “∞,” which traditionally indi-
cates the potential infinite, to “ω,” the final letter of the Greek alphabet, to
represent the infinite as a completed whole, that is, the actual infinite.
In another terminological shift, indicating his turning away from subsets
of real numbers and to abstract sets themselves, Cantor stops speaking
of point manifolds and instead talks just of sets. The major achievement of
the Grundlagen, as these changes indicate, was to single out the transfinite
numbers as both an autonomous and systematic extension of the finite
numbers. Without this move, Cantor would have had no way to progress
beyond the finite, and research in abstract set theory and on the continuum
would have come to a standstill.
In contrast with Cantor’s new approach, mathematicians had tradi-
tionally treated the infinite as a variable increasing beyond all limits or
decreasing to arbitrary smallness. In being arbitrarily large or small, how-
ever, the infinite was only potential, with the finite remaining the funda-
mental notion. The basic idea was that there was always some finite
number, large or small enough, available for the task at hand. Actual, or
proper, infinities, that is, infinities considered as complete in themselves,
were thought to be incoherent. Cantor aimed to show in the Grundlagen
that this was not the case. A succession of actually infinite numbers could
be developed with identifiable and determinate number-theoretic proper-
ties, making these transfinite numbers just as legitimate as the systems of
finite numbers. Before this work, Cantor had no simple definition of
the powers beyond the denumerable infinite. The least power among the
infinite sets was the countable set of natural numbers. His new transfi-
nite numbers would allow for a natural definition of powers beyond this
countable infinity. To this end, he observed that the natural numbers
resulted from the repeated addition of units to yield the sequence , ,
, … He calls this the first principle of generation, the successive generation
of finite ordinal numbers by successive addition. This class, the number


Georg Cantor, “Foundations of a General Theory of Manifolds: A Mathematico-Philosophical
Investigation into the Theory of the Infinite,” in From Kant to Hilbert, ed. Ewald, pp. –.

Kanamori, “The Mathematical Development of Set Theory from Cantor to Cohen,” p. ;
Dauben, Cantor, p. .

Dauben, Georg Cantor, p. .
 Quine, New Foundations, and the Philosophy of Set Theory
class (I), had no largest element, but there was nothing to stop Cantor
from introducing a new number ω expressing the natural order of the
entire set (I). This number is the first number following the entire
sequence of finite numbers in the set of natural numbers; it is the first
transfinite number. But there was also no reason not to apply the first prin-
ciple of generation again to yield additional transfinite numbers ω, ω + ,
ω + , …, ω + n, … Again, this sequence had no largest element, but
Cantor then introduced another new number, ω, to represent this entire
sequence in its natural order. He again applied the first principle of genera-
tion to yield the new sequence ω, ω + , ω + , …, ω + n, …
This process then led Cantor to characterize what he called the second
principle of generation. He explained that ω could be thought of as a limit
approached by the sequence of natural numbers but never reached by it
in the sense that ω is the first whole number after all the finite numbers
in the set of natural numbers. So his second principle of generation states
that if a sequence of numbers has no greatest element, then a new
transfinite number can always be generated as the least number greater
than all those in the sequence. So the successive application of the two
principles of generation always allows for the possibility of generating a
new number in succession to those previously generated numbers. Once
Cantor had this second principle, he was able to define his second
number class (II) as the collection of all numbers α formed from the two
generating principles in a definite increasing succession in which all num-
bers proceeding α constitute a set with the same power as the first
number class (I). He also indicated a third principle, the principle of
limitation, that was to allow him to proceed to even higher classes of
numbers, though he did not do much to develop this idea.
In summarizing the significance of this work on the first and second
number classes, Dauben notes the important difference mentioned earlier
between the transfinite numbers and Cantor’s earlier introduction of infi-
nite symbols. Previously, Cantor had focused on derived sets of the second
species and treated the transfinite symbols attached to his operator P′ as
mere indices to identify and distinguish among the derived sets themselves.
However, the transfinite numbers were themselves independent numbers,
and this was necessarily so. Cantor aimed to use these very numbers in


Dauben, Georg Cantor, pp. –. Cantor presents his three principles of generation in sections 
and  of the Foundations. We will see in Chapter  how Cantor avoided the paradoxes, the
Burali-Forti paradox being the relevant one in this case. This approach to set theory is developed
further by Michael Hallett in his Cantorian Set Theory and Limitation of Size (Oxford: Clarendon
Press, ).
 Cantor and the Early Development of Set Theory 
his further investigations into the powers of sets and the continuum. Hence
he could not define the transfinite numbers in terms of the very sets he
wished to study. They required an independent formulation so that they
could be applied to the study of point sets and their powers. The transfinite
numbers were to be understood as having an independent claim on reality
equal to that of any of the other numbers.
Another crucial aspect of the development of set theory introduced in
the Grundlagen was the well-ordering principle, which Cantor described
as “the law of thought that says that it is always possible to bring any
well-defined set into the form of a well-ordered set – a law that seems to
me fundamental and momentous and quite astonishing by reason of its
general validity.” As Kanamori points out, this principle can be under-
stood as part of the unity Cantor saw between the finite and transfinite
numbers; just as the finite numbers can be well ordered, so can the trans-
finite numbers. Cantor introduces the notion of a well-ordered set early
in the Grundlagen stating that
[a] well-ordered set is a well-defined set in which the elements are bound
to one another by a determinate given succession such that (i) there is a
first element of the set; (ii) every single element (provided it is not the last
in the succession) is followed by another determinate element; and (iii)
for any desired finite or infinite set of elements there exists a determinate
element which is their immediate successor in the succession (unless there
is absolutely nothing in the succession following all of them).
These well-ordered sets were essential to his investigations into the trans-
finite numbers and in particular, in distinguishing finite from infinite
sets. Here, he introduced the important concept of a numbering
[Anzahl], where a numbering expressed the ordering of the elements of a
given set. Later, he would identify these numberings as the ordinal num-
bers themselves. In drawing this connection between well-ordered sets
and their numberings, Cantor was also able to further the transfinite
numbers’ claim to having objective reality. This objective reality of
the transfinite numbers came from the existence of a well-ordered set
whose order could be expressed by associating them with a number from
the various transfinite number classes. To this end, Cantor aimed to
show that for any countably infinite well-ordered set, there was always a


Dauben, Georg Cantor, pp. –.

Cantor, “Foundations of a General Theory of Manifolds,” p. .

Kanamori, “The Mathematical Development of Set Theory from Cantor to Cohen,” p. .

Cantor, Foundations of a General Theory of Manifolds, p. .
 Quine, New Foundations, and the Philosophy of Set Theory
number of the second number class (II) that uniquely represented its
ordering.
To illustrate this, Cantor considered the denumerable set (αν), which
can be well ordered, for example, in any of the following ways:
ðα1 ; α2 ; …; αv ; αvþ1 ; …Þ
ðα2 ; α3 ; …; αvþ1 ; αvþ2 ; …; α1 Þ
ðα3 ; α4 ; …; αvþ2 ; αvþ3 ; …; α1 ; α2 Þ
ðα1 ; α3 ; …; α2 ; α4 ; …Þ:
Cantor states that two well-ordered sets have the same numbering, or are
similar, if they can be put into a one-to-one correspondence in such a
way that preserves their respective orderings. So if αn comes before αm in
the ordering of one set, then in the other set, the respective correspond-
ing elements αn′ and α′m must be ordered so that αn′ comes before α′m .
Such correspondences are always uniquely determined. So Cantor
observed that given any two well-ordered sets, he could use the succession
of natural numbers plus the transfinite numbers to identify similar well-
ordered sets. Given any number α of the first or second number class,
when taken together with all of its preceding elements, the numbering of
all similar well-ordered sets is given by α uniquely. For example,
ðα1 ; α2 ; α3 ; …; αv ; αvþ1 ; …Þ
ðα2 ; α1 ; α4 ; …; αvþ1 ; αv ; …Þ
ð1; 2; 3; …; v; …Þ
all have the same numbering ω. Similarly,
ðα2 ; α3 ; …; αv ; …; α1 Þ
ðα3 ; α4 ; …; αvþ1 ; …; α1 ; α2 Þ
ðα1 ; α3 ; …; ; α2 ; α4 ; …Þ
have the distinct numberings ω + , ω + , and ω, respectively.
Cantor also employed his distinction between number and numbering
to bring out differences between finite and infinite sets. For finite sets,
their numbering remains the same regardless of their ordering, whereas
for infinite sets, there are sets of the same power with different number-
ings and so with different well-orderings. The numberings, then, depend


Dauben, Georg Cantor, p. . The examples Dauben presents here and below come from the
French version of Cantor’s “Foundations of a General Theory of Manifolds;” Georg Cantor,
“Fondaments d’une Théorie Générale des Ensembles,” Acta Mathematica  (), p. .

Ibid., p. .
 Cantor and the Early Development of Set Theory 
upon the ordering of the elements so, in general, different orderings pro-
duce different numberings, although they have the same number of
elements. In addition, there is a correlation between the number of ele-
ments in a set and the number of numberings the set could yield
depending upon its ordering. For example, Cantor considered the sets of
the first number class (I) given in a determinate order. The numberings
of these sets, so long as they are well-ordered, always correspond to num-
bers of the second number class (II). Also, conversely, given any number
α of (II), any set of (I) can be ordered so that its numbering would corre-
spond to α. Analogous results hold, as well, for sets of higher power.
As was indicated previously, when focused only on finite sets, the con-
cepts of power and numbering coincide; different orderings of finite sets
do not produce different powers. Since for finite sets power was indepen-
dent of ordering, a finite set of n elements just had power n. But for
infinite sets the distinction between power and number was significant.
Every number α of (II) indicated a unique ordering of elements, but any
such set with α as its numbering was always denumerable. Here, a con-
nection remained between power and numbering in that any denumer-
able well-ordered set had a uniquely determined numbering α, where α
belonged to the second number class (II). The properties of well-ordered
sets had the result of joining these various notions of transfinite numbers,
numberings, and number classes together into a unified theory. The con-
cept of numbering merely generalized the notion of counting. For
Cantor, this further supported his view that the transfinite numbers were
just as legitimate as their finite counterparts.
Kanamori explains that Cantor’s interest in the well-ordering principle
and the ordinal numbers, that is, the numberings, was directly linked to
his attempts to solve the continuum problem. The transfinite ordinals
provided Cantor with a framework for his two primary approaches to the
problem: one through power and the other through definable sets of real
numbers. His approach through power focused on the first and second
number classes – the natural numbers and the set of countably infinite
ordinals respectively – although he also gave some indication that these
classes could be continued to a third number class and beyond. His
major result here was to prove that the number class (II) was uncounta-
ble and that any subset of (II) is either countable or of the same power


Ibid., pp. –.

Ibid., p. .

Kanamori, “The Mathematical Development of Set Theory from Cantor to Cohen,” p. .
 Quine, New Foundations, and the Philosophy of Set Theory
as (II). Thus the second number class (II) had precisely the property
Cantor sought for the real numbers, reducing the continuum hypothesis
to the assertion that the set of real numbers and the second number class
were of the same power. However, he could not seem to find a definable
well-ordering of the real numbers and so could not find a correlation
between these two sets. Cantor had not yet developed his famous diago-
nal method of proof for showing (II) uncountable. Instead, in section 
of the Grundlagen, he presented an argument similar to that given earlier
in his proof that the real numbers are uncountable.

. Cardinals, Ordinals, and the Continuum Problem

In the years following publication of the Grundlagen, Cantor continued
to be unable to solve the continuum problem by searching for direct cor-
relations between the set of real numbers and the ordinals. This led him
to a more general approach to size and order that would take into
account the continuum. To this end, he introduced the notion of a car-
dinal number to replace the earlier terminology of “power.” He also
went beyond the study of well-orderings to the more general notion of
linear order types, and he took on a view of well-defined sets as being
given together with a linear ordering of their members. Order types and
cardinal numbers resulted from the successive abstraction from a set M
to its order type M and then to its cardinal M. For ordinals the process
is similar, but we do not also abstract away the ordering of the set.
It was in this context that Cantor put forward in his  “On an
Elementary Question in the Theory of Manifolds” his famous diagonal
argument establishing that for any set M, the collection of functions
from M into a two-element set is of higher power, or cardinality, than M
itself.
As we saw, Cantor proved already in  the existence of uncounta-
ble sets. This proof, however, relied on the existence of irrational num-
bers. In , he reproved this claim with much greater generality,
appealing only to the abstract sets themselves. But in doing so, he also
employed extremely powerful methods, leading him to an ascending and
limitless hierarchy of transfinite powers. With this, he brought the


Georg Candor, Contributions to the Founding of the Theory of Transfinite Numbers (, ),
trans. Philip E.B. Jourdain (New York, NY: Dover Publications, Inc., ), pp. –.

Kanamori, “The Mathematical Development of Set Theory from Cantor to Cohen,” p. ;
Cantor, “On an Elementary Question in the Theory of Manifolds,” in From Kant to Hilbert,
Ewald, pp. –.
 Cantor and the Early Development of Set Theory 
mathematical infinite to full maturity. Cantor’s diagonal method allowed
him to avoid all talk of point sets, irrational numbers, or any specific
objects at all. In this way, he achieved complete generality, relying on
collections of no specific sorts of potentially questionable entities at all,
except perhaps his sets themselves.
The diagonalization proof relied on two elements m and w. Cantor
used these to consider a collection M made up of elements E = (x,
x, …, xn, …), where each xn was either m or w. So, for example, the ele-
ments E might look like any of the following:
E I ¼ ðm; m; m; m; …Þ;
E II ¼ ðw; w; w; w; …Þ;
E III ¼ ðm; w; m; w; …Þ:
He then claimed that any such collection M was uncountable. If E,
E, …, Eν, … is any simply infinite sequence of elements of the set M,
then there is an element E in M which is not equal to any Eν. To prove
his claim, he first gave a countable listing of elements Eμ, where each aμ,ν
was either m or w:
E1 ¼ ða11 ; a12 ; …; a1ν ; …Þ;
E2 ¼ ða21 ; a22 ; …; a2ν ; …Þ;
⋯
Eμ ¼ ðaμ1 ; aμ2 ; …; aμν ; …Þ
…
This defined a new sequence b, b, …, bν, …, where each bν was either
m or w but bν ≠ aνν. This sequence of bν yielded a new element E =
(b, b, …, bν, …) in M, where E ≠ Eν for any index ν. No matter
what element Eν was considered, E always differed from it at the νth
coordinate. Therefore there was always an element left off of a countable
listing of elements of M, and hence M was uncountable.
Cantor then proceeded to show how to apply his diagonal method to
specific infinite sets, considering the set L of the linear continuum and
the set of real numbers on (, ) and then showing that the set M of
single-valued functions f (x) with only values  or  for any x in (, )
was of greater power than L. M was clearly greater than or equal to L in


Dauben, Georg Cantor, p. ; Cantor, “On an Elementary Question in the Theory of
Manifolds,” pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
power, since it contained a subset equal in power to L. For example, the
set N of functions f (x) on (, ) equal to  everywhere except at a single
point x, where f (x) =  was such a subset. So Cantor had to show only
that L and M are in fact not equal in power. If M and L are equal in
power, then it must be possible to establish a one-to-one mapping
between them. That is, there must be a function φ(x, z), where for every
value z, there must be an element f (x) in M such that f (x) = φ(x, z).
Conversely, for every element f (x) in M given by φ(x, z), there must be a
unique z such that f (x) = φ(x, z). This is impossible, as is shown by the
diagonalization method. Consider the function g(x) having only values 
or  but where g(x) ≠ φ(x, x) for any given x. Hence g(x) is in M, but z
could not be determined in a way that would yield g(x) from φ(x, z),
since φ(z, z) is never equal to g(z). Therefore M is of greater power
than L.
From his new proof of the existence of uncountable sets, Cantor pro-
vided a simple way for showing that the ascending sequence of powers of
well-defined sets had no maximum. Indeed, he took this to be the true
significance of the proof:
This proof is remarkable not only because of its great simplicity, but more
importantly because the principle therein can be extended immediately to
the general theorem that the powers of well-defined manifolds have no
maximum, or what is the same thing, that for any given manifold L we
can produce a manifold M whose power is greater than that of L.
Furthermore, Cantor took this infinite ascending sequence of higher and
higher powers as a generalization of the notion of a finite cardinal and
therefore as having a status equal to that of the finite numbers. As he
remarked,
The “powers” represent the unique and necessary generalization of the
finite “cardinal numbers.” They are none other than the actual-infinite
cardinal numbers, and they have the same reality and determinateness as
the others, except that the law-like relations among them – their “number
theory” – is in part of a different sort than in the domain of the finite.
For Cantor – looking at the history of mathematics – the rational, irra-
tional, and complex numbers had been accepted because of their utility
and consistency. Each of these number systems was a consistent

Dauben, Georg Cantor, p. ; Cantor, “On an Elementary Question in the Theory of
Manifolds,” p. .

Ibid., pp. –; Dauben also quotes this passage on p.  of Georg Cantor.

Cantor, “On an Elementary Question in the Theory of Manifolds,” p. .
 Cantor and the Early Development of Set Theory 
generalization of prior less comprehensive concepts. The concept of
power, then, was the most comprehensive and natural of all.
Cantor’s final major work in set theory was his /
Contributions to the Founding of the Theory of Transfinite Numbers, com-
monly known as the Beiträge. This work summarizes his progress in set
theory to this point while also clearly revealing a gap left open by his
inability to settle the continuum problem. In part I, he presents his post-
Grundlagen research on cardinal numbers and the continuum, posing the
question of cardinal comparability, whether for all cardinals a and b,
either a = b, a < b, or b < a. He promised a proof at some point later
but never provided one, and it did not come until it appeared as a conse-
quence of Zermelo’s  well-ordering theorem. Putting comparability
aside for now, however, Cantor first turned to defining the addition,
multiplication, and exponentiation of cardinal numbers, showing that,
like the finite numbers, the infinite too were governed by arithmetical
laws. He also introduced the now standard aleph notation for infinite
cardinals, taking ℵ as the cardinality of the set of natural numbers, the
first infinite cardinal number. He further observed that 2ℵ0 is the cardin-
ality of the set of real numbers and easily established, by way of his cardi-
nal arithmetic, the one-to-one correspondence between R and Rn, since
ð2ℵ0 Þℵ0 ¼ 2ℵ0 ⋅ ℵ0 ¼ 2ℵ0 . Finally, he presented his theory of order types,
taking η as the order type of the rational numbers, a countable dense lin-
ear order without endpoints, and θ as the order type of the real numbers,
a perfect linear order with a countable dense set.
In part II of the Beiträge, Cantor further developed his views on well-
orderings construing their order types as the ordinals and proving, by way
of order comparisons of well-ordered sets, that they are comparable. He
then presented ordinal arithmetic as just a special case of the arithmetic of
order types. He also paid some attention to the properties of the second
number class (II) and defined its cardinal number as ℵ. In the final sec-
tions, he turned to ordinal exponentiation in the second number class,
defining this operation by transfinite recursion and using it to establish his
famous normal form theorem. He concluded with a discussion of the num-
bers satisfying the condition ε = ωε, now known as the epsilon numbers.


Dauben, Georg Cantor, p. .

Georg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers, trans. Philip
E. B. Jourdain (New York, NY: Dover Publications, ).

Kanamori, “The Mathematical Development of Set Theory from Cantor to Cohen,” p. ;
Cantor, Contributions, pp. , –, , .

Ibid., p. ; Cantor, Contributions, pp. , –.
 Quine, New Foundations, and the Philosophy of Set Theory
With the first part of the Beiträge dealing primarily with cardinal
numbers and the continuum and the second part focusing on ordinal
numbers and well-orderings, the two parts remained distinct and lacking
in unity in their subject matter. Kanamori states that this was not just an
oversight but the indication of a serious split between the areas of
Cantor’s developing set theory. For example, nowhere in part I does
Cantor state, even as a special case, his major  result that a < a for
any set a, either finite or infinite. Instead, the succession of transfinite
cardinals is taken as the alephs defined as the powers of the sets of their
predecessor ordinals. But then in part II, Cantor never mentions any
aleph beyond ℵ, or that the continuum hypothesis can be stated as
2ℵ0 ¼ ℵ1 . Furthermore, he establishes ordinal comparability but is
unable to reduce cardinal comparability to it. Kanamori concludes,
“Having ushered in arbitrary functions through cardinal exponentiation
Cantor had introduced an irreconcilable tension into his view that all
sets are well-ordered, and there was little point to developing the theory
of higher alephs with the assurance of their gauging all the cardinal num-
bers.” However, the  International Congress of Mathematicians at
Heidelberg signaled a turning point. Julius König presented a proof of
the claim that 2ℵ0 was not an aleph, meaning that the continuum was in
fact not well-orderable. Although by the next day, Zermelo had shown
König’s proof to be faulty, Cantor was shaken with worries that the con-
tinuum might escape the context for quantifying it that he had worked
to construct for this very purpose. This was not the only problem
looming for Cantor’s set theory. By this time, the paradoxes of set theory
were also threatening, and it is to this topic that I turn next.


Ibid., p. .

Ibid., pp. –.

Ibid., p. .
 CHAPTER 

Cantor, Russell, and Zermelo and

the Set-Theoretic Paradoxes

In this chapter, I turn to consider the set-theoretic paradoxes in their his-

torical and philosophical context. There has been much recent scholar-
ship in this area, attempting to sort out the rather complicated history of
how the paradoxes arose, when they were discovered, and by whom.
Earlier accounts of this history had Cesare Burali-Forti discovering the
first of the paradoxes – that of the largest ordinal, which became the
Burali-Forti paradox – with the Cantor and Russell paradoxes following
soon after. More recently, this story has seen considerable revision, par-
ticularly at the hands of Gregory Moore and Alejandro Garciadiego.
Specifically, they have argued that the discovery of the earliest set-
theoretic paradoxes – Burali-Forti’s, Georg Cantor’s, and Bertrand
Russell’s – can all be traced to Russell himself. Following them, I pick
up the story of the paradoxes with Russell.
In tracing the emergence of the Burali-Forti paradox to Russell,
Moore and Garciadiego point to Russell’s early Hegelianism as its source,
explaining, “Russell began to search for paradoxes in mathematics much
earlier than is usually recognized. His predisposition to invent such para-
doxes had its roots in the philosophical antinomies of Kant and Hegel,


For example, see Jean van Heijenoort’s introductory note to Cesare Burali-Forti, “A Question on
Transfinite Numbers,” in From Frege to Gödel: A Source Book in Mathematical Logic, –,
ed. Jean van Heijenoort (Cambridge, MA: Harvard University Press, ), pp. –.

Gregory H. Moore and Alejandro Garciadiego, “Burali-Forti’s Paradox: A Reappraisal of Its
Origins,” Historia Mathematica  (), pp. –; Alejandro Garciadiego, Bertrand Russell and
the Origins of the Set-Theoretic “Paradoxes” (Boston, MA: Birkhäuser Verlag, ). They note that
van Heijenoort was far from the only historian to hold this view, citing, among others, the similar
attitudes of I. M. Bochenski, Nicolas Bourbaki, Morris Kline, and William and Martha Kneale; see
p.  of Moore and Garciadiego, “Burali-Forti’s Paradox,” and pp. – of Garciadiego,
Bertrand Russell and the Origins of the Set-Theoretic “Paradoxes.”

The last of these needs some qualification. Ernst Zermelo had discovered Russell’s paradox inde-
pendently – and before Russell himself – but did not publish it. For more on Zermelo’s discovery,
see B. Rang and W. Thomas, “Zermelo’s Discovery of the ‘Russell Paradox.’” Historia
Mathematica  (), pp. –.


 Quine, New Foundations, and the Philosophy of Set Theory
both of whom deeply influenced his early development as a philoso-
pher.” Adding to this story, I would emphasize that Russell came upon
the set-theoretic paradoxes, including the one that would bear his name,
by considering some fairly intuitive but unfortunately contradictory fea-
tures that we might expect the notion of a set to have. This aspect in his
discovery of the paradoxes will be especially important throughout much
of what follows in this and later chapters. It brings out that the founda-
tional crisis in mathematics was a battle of intuitions about sets and that
its resolution ultimately came down to largely pragmatic considerations
which, I will claim, reveals a distinct tradition of explication in the philo-
sophy of set theory. The paradoxes showed that a single notion of set,
incorporating all of the intuitive features that both mathematical and
philosophical logicians initially thought sets to have, was impossible. As a
result, set theory’s founders looked to the mathematically significant fea-
tures of set theory that were traditionally conceived to give an explicative
account of the notion of set rather than a uniquely correct conceptual
analysis of the entities that set theory was supposed to be about.

. Russell and the Discovery of the Paradoxes

Russell had arrived at his paradox by May , which he discovered,
according to his account in Principles of Mathematics, “in the endeavor to
reconcile Cantor’s proof that there can be no greatest cardinal number
with the very plausible supposition that the class of all terms … has
necessarily the greatest possible number of members,” a term being
“[w]hatever may be an object of thought, or may occur in any true or
false proposition, or can be counted as one.” Moore puts forth this date
for the discovery based upon an early manuscript of Principles where in
chapter three, Russell composes the first extant account of his paradox:
The axiom that all referents with respect to a given relation form a class
seems, however, to require some limitation, and that for the following rea-
son. We saw that some predicates can be predicated of themselves.
Consider now those (and they are the vast majority) of which this is not
the case. These are the referents (and also the relata) in a certain complex
relation, namely the combination of non-predicability with identity.


Moore and Garciadiego, “Burali-Forti’s Paradox,” p. .

Bertrand Russell, Principles of Mathematics, nd edn (London: George Allen and Unwin, ),
p. . The first edition of Principles of Mathematics appeared in . Russell defines the terms
on p. . He makes the same claim in his later reminisces; see Bertrand Russell, My Philosophical
Development (New York, NY: Routledge,  []), p. .
 Cantor, Russell, and Zermelo and the Set-Theoretic Paradoxes 
But there is no predicate which attaches to all of them and to no other
terms. For this predicate will either be predicable or not predicable of
itself. If it is predicable of itself, it is one of those referents by relation to
which it was defined, and therefore, in virtue of their definition, it is not
predicable of itself. Conversely, if it is not predicable of itself, then again
it is one of the said referents, of all of which (by hypothesis), it is predic-
able, and therefore again it is predicable of itself. This is a contradiction,
which shows that all the referents considered have no common predicate
and therefore do not form a class … It follows from the above that not
every definable collection of terms forms a class defined by a common
predicate. This fact must be borne in mind, and we must endeavor to dis-
cover what properties a collection must have in order to form a class.
Here, we see not only Russell’s first statement of the paradox, but he
also gives the first indication of where its resolution will ultimately lie. In
observing that “the axiom that all referents with respect to a given rela-
tion form a class” needs limiting, Russell suggests the traditional concep-
tion of what a class is: that every predicate or concept determines a class.
As we have seen and will continue to see, this idea of a set was found
throughout the emerging discipline of set theory, including that work of
Cantor. Locating the source of the paradox in the comprehension princi-
ple changed the development of set theory in crucial ways and, in parti-
cular, gave rise to the philosophical debates over set theory that we will
primarily be concerned with in many of the following chapters. I should
also note that in moving toward rejecting full comprehension, Russell
does not completely reject the idea that there is a uniquely correct ver-
sion of set in favor of a more pragmatic conception, that is, Russell does
not completely move from the conceptual analysis view to the explication
view. (I do not think Russell ever adopts one view or the other comple-
tely, which often leads to tensions in his philosophy of mathematics.)
Instead, as Russell remarks at the end of the passage, now the task
becomes “to discover what properties a collection must have in order to
form a class.” He has not yet suggested that the mathematical demands
of set theory can settle this question.
Russell did not, however, react with immediate concern. The available
textual evidence suggests that he reported the paradox to no one else –
except Giuseppe Peano, who never responded – until June , when

Bertrand Russell, The Collected Papers of Bertrand Russell, vol.  (New York, NY: Routledge,
), p. . A version of this passage is quoted in Gregory H. Moore, “The Origins of Russell’s
Paradox: Russell, Couturat, and the Antinomy of Infinite Number,” in Essays on the Development
of the Foundations of Mathematics, ed. Jaakko Hintikka (Dordrecht: Kluwer Academic Publishers,
), pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
he communicated it to Gottlob Frege. At that point, Frege took it as a
matter of great concern:
Your discovery of the contradiction caused me the greatest surprise and,
I would almost say, consternation, since it has shaken the basis on which
I intended to build arithmetic. It seems, then, that transforming the gen-
eralization of an equality into an equality of courses-of-values (§  of my
Grundgesetze) is not always permitted, that my Rule V (§ , p. ) is
false, and that my explanations in §  are not sufficient to ensure that
my combinations of signs have a meaning in all cases. I must reflect
further on the matter. It is all the more serious since, with the loss of my
Rule V, not only the foundation of my arithmetic, but also the sole possi-
ble foundations of arithmetic, seem to vanish.
Moore reasonably suggests that Russell was not initially disturbed by his
paradox because he thought there must be some simple solution to it. Only
upon receiving Frege’s pessimistic response did Russell see how devastating
the consequences of his discovery were. He looked upon Frege with extre-
mely high regard, and when he could offer no simple resolution, Russell
could no longer see this as a minor difficulty that would be solved in time.
For Russell, the foundational crisis had arrived.
From this point, matters became worse. Moore and Garciadiego point
to Russell’s recognition of the seriousness of his own paradox as the
source of the other set-theoretic paradoxes. Now, reconsidering the work
of Burali-Forti and of Cantor, Russell came to recognize also the two
paradoxes that would eventually bear their respective names. In a 
article on well-ordering, Russell began his attempts at reconciling the
works of Burali-Forti and Cantor, now accepting Cantor’s proof of
the comparability of ordinals but doubting his claim that every set can
be well ordered. Russell’s response at this point was to grant that every
ordinal segment was well-orderable but not that the ordinals as a whole
were. In this way, he continued to avoid the conclusion that Burali-Forti
and Cantor had landed themselves in contradiction.
This situation changed in the  publication of Principles of
Mathematics. Here, in addition to an entire chapter on his own paradox,
Russell put forward, for the first time in print, both the Burali-Forti and


Gottlob Frege, “Letter to Russell” (), in From Frege to Gödel, ed. van Heijenoort, pp. –.
I have followed the translation given there. The “Rule V” that Frege refers to is what is more com-
monly known as his “Basic Law V.”

G. H. Moore, “The Origins of Russell’s Paradox,” pp. –; see also Moore and Garciadiego,
“Burali-Forti’s Paradox,” pp. –.

Moore and Garciadiego, “Burali-Forti’s Paradox,” pp. –.
 Cantor, Russell, and Zermelo and the Set-Theoretic Paradoxes 
Cantor paradoxes. As already mentioned previously, Russell claims to
have discovered his own paradox in considering Cantor’s theorem. In
chapter ten of Principles of Mathematics, entitled “The Contradiction,”
he gives extensive treatment to the paradoxes, rehearsing his own in three
different forms. In diagnosing the problem, he again turns to the com-
prehension principle, the axiom that any propositional function in one
free variable determines a class. He observes that either this principle or
the principle that every class can be treated as a single term must be false,
but he sees no fundamental objection to rejecting either. “But having
dropped the former,” Russell asks, “[w]hich propositional functions
define classes which are single terms as well as many, and which do not?
And with this question our real difficulties begin [my emphasis].” This, as
we will continue to see, is indeed the crucial issue to set theory in the
early twentieth century and, in a sense, remains to this day one of the
crucial issues for set theory in its more philosophical aspects. Having
spelled out his own paradox, Russell now returns to Burali-Forti and
Cantor, seeing for the first time that their views also lead him into diffi-
culty, though of a less general nature than the one that emerges from his
own paradox.
Russell first considers Burali-Forti, remarking that there appears to be
a problem with the order type of the whole series of ordinals. We can
easily prove that every segment of this series is well ordered, so it is nat-
ural to suppose that the whole series is also well ordered. The order type
of this series would be the greatest of all the ordinals, since the ordinals
that are less than a given ordinal form, in order of increasing magnitude,
a series whose order type is the given ordinal. But every ordinal can be
increased by the addition of , so there can be no greatest ordinal. On
the basis of this contradiction, Russell correctly observes that Burali-Forti
concluded that for any two given ordinal numbers, comparability does
not in general hold. Russell also observes that Cantor proved exactly the
opposite. Here, we have Russell’s discovery of the Burali-Forti paradox.
Russell finds no weakness in Cantor’s contrary proof. Instead, he ques-
tions Burali-Forti’s premise that the whole series of ordinals is well
ordered. From the well-ordering of every segment of ordinals, Russell


Some scholars have traced the Cantor paradox to the  Cantor–Dedekind correspondence,
but as we will see, this is not entirely accurate. Cantor does produce a contradiction there but,
much like Burali-Forti, only in the service of a reductio proof. It was Russell who turned these
contradictions into a more general foundational crisis for mathematics.

Russell, Principles of Mathematics, pp. –.

Ibid., p. .
 Quine, New Foundations, and the Philosophy of Set Theory
claims that the well-ordering of the whole series does not follow. Because
this claim of the well-ordering of the entire series of ordinals appears
incapable of proof and rejecting it avoids Burali-Forti’s paradox, Russell
follows this path and, at least for now, takes this to be the successful
resolution of the paradox of the largest ordinal. Russell does not see,
however, that Cantor himself nowhere explicitly admits a class of all ordi-
nals, and that without this, there is no contradiction. What drives
Russell to the paradox is his perhaps not unreasonable view that the ordi-
nals themselves should form a class. Had there been no paradox, it seems
no more surprising to have a class of all ordinals than to have a class of
all natural or real numbers.
In introducing his final set-theoretic paradox, Russell turns to Cantor
himself, considering his investigations into the cardinal numbers. Russell
begins by remarking that the common objections to infinite numbers,
classes, series, and the infinite in general as self-contradictory are ground-
less, but more serious problems remain, related to his own paradox of
non-self-membered classes. This is not a problem with the infinite as
such, he argues, but only with certain large infinite classes, explaining the
difficulty as follows. Cantor proved that there is no greatest cardinal
number: If u is a class, then the number of classes included in u is greater
than the number of terms, or elements, in u. In other words, for any car-
dinal α, α > α, which is Cantor’s theorem. But Russell observes also
that certain classes appear to have as many terms as possible, such as the
class of all terms, the class of all classes, and the class of all propositions.
So it appears to Russell that Cantor’s proof must make some assumption
that does not hold for such classes, since definite contradictions arise in
applying Cantor’s reasoning to these large classes. Indeed, such difficul-
ties arise generally for any class of all entities or any equally numerous
class. Russell admits that in light of this problem, we might be tempted
to think that the totality of things, or the whole universe of existing enti-
ties, is an illegitimate totality, inherently contradictory to logic. “But it is
undesirable to adopt so desperate a measure,” he counters, “so long as
hope remains of some less heroic solution.”
Russell concludes these investigations by observing that although
Cantor’s argument leads to contradictions where such large classes are
concerned, he can find no missteps in Cantor’s original proof. The only
solution he sees at this point is to deny that there are any true


Ibid.

Ibid., p. .
 Cantor, Russell, and Zermelo and the Set-Theoretic Paradoxes 
propositions concerning all objects or propositions. “Yet the latter, at
least, seems plainly false,” he says, “since all propositions are at any rate
true or false, even if they have no other common properties. In this unsa-
tisfactory state, I reluctantly leave the problem to the ingenuity of the
reader.” For Cantor, by contrast, we will see that it seems plainly true
that there is no universal class. I will now consider how these difficulties
were in fact dealt with by Russell and others.

. Avoiding the Paradoxes: Cantor and the Absolute Infinite

I turn first to Cantor, both as the founder of set theory and as someone
who may never have been confronted by problems such as those that
Russell introduced. For while Russell saw it as near common sense that
such large classes as the class of all classes and the class of all objects
should exist, Cantor saw this as equally implausible. As we will see, in
Cantor’s view of the nature classes, we are unable to derive such para-
doxes. The only contradictions Cantor derives are the sorts one would
ordinarily and unsurprisingly expect to find in reductio proofs. The
first occurrence of Cantor’s paradox has often been traced to his  corre-
spondence with Richard Dedekind, but this interpretation presents two
problems: the one just mentioned and the other that Cantor’s paradox was
not publicly known before  when the relevant pieces of the Cantor–
Dedekind correspondence were published. There are, however, still interest-
ing and relevant aspects of this correspondence for the set-theoretic
paradoxes, in particular for understanding how they were resolved.
Cantor initiated his  correspondence with Dedekind at the end
of July, writing that he would like them to correspond regularly so that
he might have Dedekind’s views on certain fundamental questions of set
theory. In particular, Cantor wrote to Dedekind on August , present-
ing his proof that every transfinite set has a definite cardinal and identify-
ing the system л (tav) of all alephs with the system of all transfinite
cardinal numbers, that is, all transfinite cardinals are alephs. But before
proving this result, Cantor introduces what he sees as a crucial

Ibid., p. .

This is as close as Burali-Forti ever came to paradox as well, as is shown in his “A Question on
Transfinite Numbers,” pp. –. Here, he does not show that Cantor’s theory is inconsistent
but, rather, attempts to prove the trichotomy law for ordinals by way of a reductio proof. There is
a contradiction in the paper but only the one needed for the reductio.

Georg Cantor, Richard Dedekind, and David Hilbert, “Cantor’s Late Correspondence with
Dedekind and Hilbert,” in From Kant to Hilbert,: A Source Book in the Foundations of
Mathematics, vol. II, ed. William Ewald (Oxford: Clarendon Press, ), pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
distinction: that between consistent and inconsistent multiplicities. He
explains that he begins with the notion of a definite multiplicity (or sys-
tem or totality) of things but then observes that the assumption that a
multiplicity’s elements “are together” can lead to a contradiction in cer-
tain cases. So contrary to our original thought, such a multiplicity cannot
be conceived of as a unity, or “one finished thing,” and these multiplici-
ties, he says, are the absolutely infinite, or inconsistent, multiplicities. He
observes that “[a]s we can readily see, the ‘totality of everything think-
able’, for example, is such a multiplicity; later still other examples will
turn up.” Of course, as we saw, Russell did not readily see this, and his
initial commitment to such classes is what brought him to the paradoxes.
The idea that I will begin to develop here and advance more in future
chapters is that the systemization of set theory has largely been, and
remains, a battle of competing intuitions about what sets are like. As we
will see, what set theory’s founders needed was a principled way for
determining which sets exist that would also eliminate the paradoxes.
In contrast to his inconsistent multiplicities, Cantor also observes that
there are collections in which the elements can be thought as “being
together” without contradiction and so can be gathered into “one thing.”
These are the consistent multiplicities, or sets. He goes on to note that if
two multiplicities are equivalent, then they are both either sets or incon-
sistent and every submultiplicity of a set is also a set. Also, for any set of
sets, the elements of these sets also form a set. Using this distinction
between consistent and inconsistent multiplicities, Cantor begins his
proof that every transfinite cardinal is an aleph. First, he considers the
system Ω of all ordinals, noting that he had earlier proved that any two
ordinals are comparable. This multiplicity, when naturally ordered
according to magnitude, forms a well-ordered sequence
0; 1; 2; 3; …; ω0 ; ω0 þ 1; …; γ; …
in which every number is the order type of the preceding sequence of
elements. Next, he shows that Ω is not a consistent multiplicity. Assume
that Ω is consistent. Then, since it is well ordered, it corresponds to
some ordinal δ greater than all the ordinals in Ω. But δ is also in Ω,
since Ω contains all the ordinals. So we have δ < δ, a contradiction.
Therefore Ω is an inconsistent, or absolutely infinite, multiplicity.

Ibid., pp. –.

Ibid., p. .

Cantor actually considers Ω′, which he takes as the ordinals plus zero. For simplicity, I will just
take the ordinals to include zero.
 Cantor, Russell, and Zermelo and the Set-Theoretic Paradoxes 
Cantor then further explains that since the similarity of well-ordered
sets also yields their equivalence – that is, they have the same cardinality –
to every ordinal γ there corresponds a definite cardinal c ¼ γ where γ is
the general notion applying to all ordered sets similar to γ. That is, c is the
cardinal of any well-ordered set with type γ. As we saw, Cantor denotes the
cardinals corresponding to the transfinite ordinals by “ℵn,” aleph, the first
letter of the Hebrew alphabet. He explains to Dedekind that the system of
ordinals γ corresponding to one and the same cardinal forms “a number
class,” Z(c). In every such number class, there is a least ordinal γ, and
there is also a first ordinal γ falling outside this number class Z(c) such
that the condition γ < γ < γ is equivalent to the fact that γ belongs to
Z(c). Therefore every number class is a definite segment of the sequence
Ω. Finally, Cantor notes that certain ordinals of the system Ω each by
itself forms a number class. For example, to the finite ordinals
0; 1; 2; 3; …; ν; …;
there correspond the finite cardinals
0; 1; 2; 3; …; ν; …
Now let ω be the least transfinite ordinal, and let ℵ be the cardinal
corresponding to it, so that ℵ0 = ω0 . So ℵ is the least aleph and deter-
mines the number class Z(ℵ) = Ω. The ordinals α of Ω satisfy the
condition ω < α < ω and are characterized by it, and ω, then, is the
least transfinite ordinal whose cardinal does not equal ℵ. So ω = ℵ,
and ℵ ≠ ℵ, the next greater aleph, and so on. Cantor concludes by
observing that among the transfinite ordinals of Ω, there is a least ordinal
to which no ℵν corresponds, where ν is finite, which he denotes as
“ωω0 ” This yields an aleph ℵω0 which is the next greatest cardinal after
all the ℵν. This process of forming alephs and number classes of Ω “that
correspond to them is absolutely limitless” [Cantor’s italics]. The system
of all alephs he denotes by “л,” tav, the twenty-second letter of the
Hebrew alphabet, and he can now show that it too is an inconsistent, or
absolutely, infinite totality. He observes that the system л of all alephs
that are ordered according to magnitude,
ℵ0 ; ℵ1 ; …; ℵω0 ; ℵω0þ1 ; …; ℵω1 ; …;
forms a sequence similar to the system Ω and so is also inconsistent.


Cantor, Dedekind, and Hilbert, “Cantor’s Late Correspondence,” pp. –.

Ibid., p. .
 Quine, New Foundations, and the Philosophy of Set Theory
Cantor’s final result in this letter answers the question “Are there
transfinite cardinals not contained in the system л?” He answers nega-
tively on the basis of the inconsistency of Ω and л. If we define a multi-
plicity V and assume that no cardinal corresponds to it, then V must be
inconsistent. By this assumption, Ω must be projectible into V, so there
must be a submultiplicity V′ of V equivalent to Ω. V′, then, is inconsis-
tent, since Ω is, so the same must hold of V. Therefore, he concludes,
every consistent transfinite multiplicity must have a definite aleph as its
cardinal, so all transfinite cardinals are contained in the system л. From
this, he also concludes that his  assertion of the comparability of
cardinals was in fact correct.
In an editorial note to this letter, Ernst Zermelo observes a weakness
in Cantor’s proof, namely, that Cantor had not proved that Ω was in
fact projectible into every V lacking an aleph as its cardinal. Something
like an intuition of time seemed to be at play here, with Cantor continu-
ing a process of successively and arbitrarily matching elements of V with
the ordinals of Ω. However, this assumes that each of the elements of V
will be used only once in the process, and such a process, Zermelo
remarks, “goes beyond all intuition.” What Cantor needed was
Zermelo’s axiom of choice so that he could make a simultaneous choice
of elements from V to define V′. It was this axiom that eventually
allowed Zermelo to prove what Cantor had wanted.
Although we will return to the axiom of choice in later chapters, this
weakness in Cantor’s proof does not obscure my main concern here.
What is important to note is that while something like Cantor’s paradox
does appear in this letter to Dedekind, it is all part of a reductio proof.
Cantor’s paradox did not result in a foundational crisis because this con-
tradiction was what Cantor intended to prove for reductio so that he
could obtain his results about the transfinite cardinals. Why, then, did
Russell react so differently to the paradoxes? This is the second important
point in Cantor’s letter. In his distinction between consistent and incon-
sistent totalities, Cantor had a response to the paradoxes before any foun-
dational crisis arose. But what were the grounds for such a distinction? It
is not exactly clear when Cantor introduced it, although the earliest

Ibid., pp. –.

Ibid., p. , fn. b.

Ernst Zermelo, “Proof That Every Set Can Be Well-Ordered” () in From Frege to Gödel, ed.
van Heijenoort, pp. –. Owing to controversy over his use of the axiom of choice, Zermelo
reproved the result, again using AC, in ; see “A New Proof of the Possibility of a Well-
Ordering,” in From Frege to Gödel, ed. van Heijenoort, pp. –.

Moore and Garciadiego emphasize this point in their work.
 Cantor, Russell, and Zermelo and the Set-Theoretic Paradoxes 
evidence for some version of it is an  letter to David Hilbert. One
possibility is that Cantor did in fact first discover the set-theoretic para-
doxes and then introduced it as a solution to them. If this is the case,
Cantor’s solution to the paradoxes was very much on a par with the solu-
tions offered later by Russell and Zermelo. As we will see, they were all
in a sense introduced on pragmatic grounds with the aim of preserving
as much of the mathematical content of Cantor’s original theory as possi-
ble. There does not seem to be much evidence for this as a direct
account of Cantor’s thinking on the paradoxes. However, this way of
thinking about resolving the paradoxes is consistent with the mathemati-
cal attitude toward developing set theory that we examined in Chapter .
There, we saw Cantor developing his theory in response to his mathema-
tical aims. In a sense, we will see both Zermelo and Russell adopting this
attitude in the service of coming to terms with the paradoxes themselves.
Another possibility for grounding Cantor’s appeal to consistent and
inconsistent totalities – and one that would be much more welcome to
set theorists who think there is a single correct, or intended, notion of a
set – is that Cantor in fact thought there was something inherent in
what a set is that called for such a distinction. There does seem to be
some evidence for this view. But it will not be especially satisfying to
most set theorists. Joseph Dauben has shown that Cantor worried greatly
over whether his views of the infinite were consistent with Catholic
dogma – whether knowledge of the infinite was to claim the kind of
knowledge that only God could have. The acknowledgment of absolutely
infinite totalities alleviated all such worries, since their inconsistency
showed that we did not in fact have knowledge of such infinities.
Human understanding was limited to infinities that fell short of Ω or л
or the system of all things thinkable.


There is some earlier evidence for the distinction in Cantor’s discussion of the Absolute; see for
example Cantor, Foundations, pp. –, fn. . I believe, though, that the  letter may be
the earliest evidence of Cantor’s connecting the distinction explicitly to the set-theoretic
paradoxes.

Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Princeton,
NJ: Princeton University Press, ), Chapters  and . Dauben provides various references
for Cantor’s philosophico-theological views of set theory, but for his published views, see espe-
cially “Mitteilungen zur Lehre vom Transfiniten” (, ). In Gesammelte Abhandlungen:
Mathematische und Philosophischen Inhalts, ed. Ernst Zermelo (Hildesheim: Georg Olms
Verlagsbuchhandlung, ), pp. –. Hallett, in his Cantorian Set Theory and Limitation of
Size (Oxford: Clarendon Press, ), also discusses the relationship between Cantor’s philosophy
of set theory and his theological views; see especially pp. – and –.
 Quine, New Foundations, and the Philosophy of Set Theory
While the possibly ad hoc nature of such a distinction between consis-
tent and inconsistent totalities did not bother the likes of Hilbert and
Dedekind, the unclarity of it did. As was noted previously, the earliest
explicit mention of the distinction came from a letter of September ,
, to Hilbert communicating an early version of Cantor’s proof that
all transfinite cardinals are alephs. Just as in his letter to Dedekind,
Cantor then uses a reductio proof to show that all transfinite cardinals
are alephs. While we do not have Hilbert’s responses to Cantor, Cantor’s
letter of October , , suggests that Hilbert did not fully grasp the
distinction Cantor wished to make. Cantor begins this letter by immedi-
ately observing that Hilbert overlooked the characteristic of being a
finished set. He goes on to again explain his proof, remarking:
One must only understand the expression “finished” correctly. I say of a
set that it can be thought of as finished (and call such a set, if it contains
infinitely many elements, “transfinite” or “super-finite”) if it is possible
without contradiction (as can be done with finite sets) to think of all its
elements as existing together, and so to think of the set itself as a com-
pounded thing for itself; or (in other words) if it is possible to imagine the
set as actually existing with the totality of its elements [Cantor’s italics].
He then notes that “the ‘transfinite’ coincides with what has since anti-
quity been called the ‘actual infinite.’” He explains that what he
intends by a set is “an ‘assembling together’ [which] is only possible if an
‘existing together’ is possible [Cantor’s italics].” He concludes by contrast-
ing this with the absolutely infinite sets:
Infinite sets such that the totality of their elements cannot be thought of
as “existing together” or as a “thing for itself” … and that therefore also in
this totality are absolutely not an object of further mathematical contem-
plation, I call “absolutely infinite sets”, and to them belongs the “set of all
Alephs” [Cantor’s italics].
We do not know whether Hilbert ever understood Cantor’s distinction,
but we do know that Dedekind was equally puzzled by it. In his response
to Cantor’s proof that every transfinite cardinal was an aleph, Dedekind
expressed his confusion:
You will certainly sympathize with me if I frankly confess that, although
I have read through your letter of  August many times, I am utterly


Georg Cantor, Richard Dedekind, and David Hilbert, “Cantor’s Late Correspondence with
Dedekind and Hilbert,” In Ewald, From Kant to Hilbert, pp. –.

Ibid., p. .
 Cantor, Russell, and Zermelo and the Set-Theoretic Paradoxes 
unclear about your distinction into consistent and inconsistent; I do not
know what you mean by the “coexistence of all elements of a multiplicity”,
and what you mean by its opposite. I do not doubt that with a more thor-
ough study of your letter a light will go on for me; for I have great trust in
your deep and perceptive research.
Cantor was not afraid to bring certain mentalistic elements into his set
theory, but the precise mathematical minds of Dedekind and Hilbert
were unable to comprehend what exactly Cantor’s informal distinction
came to, that is, how it was to yield objects precise enough for mathe-
matical contemplation. There seem to be both physicalistic and men-
talistic aspects of Cantor’s view of a set. He speaks of collecting
together elements into a set as a sort of pseudo-physical process that
works much like the collecting together of any other group of physical
objects. So long as we can imagine such a process, we have an existing
together of elements and so a consistent totality, or set. However, a
group of physical objects existing together is much easier to make sense
of than, say, the cardinal or ordinal numbers. For physical objects, we
might just stand them all together, and then we see that they can in
fact exist simultaneously together. Having actually done just such a
thing many times, we do not find it hard to imagine such a process
when we literally cannot bring these objects all together to one place.
Why, then, would something analogous fail for all of the cardinals? It
is not immediately obvious why we cannot imagine these objects exist-
ing together, as Cantor says that we cannot. The set of all cardinal
numbers turns out inconsistent in Cantor’s view, so in a sense maybe
we cannot think of these objects as existing together. But it is unclear
precisely why the analogy with other sorts of objects breaks down. For
Dedekind, Hilbert, and Russell, who were not already inclined toward
Cantor’s distinction, the analogy of collecting things together in the
mind did not help. What was needed was some mathematically precise
and principled way of determining which sets exist that would also
banish the paradoxes.
Looking at Cantor’s published views on the notion of a set yield no
further information as to what principle decided between consistent and


Ibid., p. .

For a defense of Cantor’s views, see William Tait, “Frege Versus Cantor and Dedekind: On the
Concept of Number,” in The Provenance of Pure Reason: Essays in the Philosophy of Mathematics
and Its History (New York, NY: Oxford University Press, ), pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
inconsistent totalities. In his Contributions to the Founding of the Theory
of Transfinite Numbers published in , Cantor defines a set as “any
collection into a whole M of definite and separate objects m of our intui-
tion or our thought.” Thus he relies on an apparently mentalistic col-
lecting together. Earlier, in a footnote to the Grundlagen of , Cantor
did elaborate some this idea of a collecting together:
Theory of manifolds. I use this word to designate a very broad theoretical
concept which I have hitherto used only in the special form of a theory of
geometric or arithmetical sets. In general, by a “manifold” or “set”
I understand every multiplicity which can be thought of as one, i.e. every
aggregate of determinate elements which can be united into a whole by
some law.
Here again, Cantor appeals to an ability of our thought as determining
which sets exist. But he does give us something further in this passage in
stating that the sets are those aggregates whose “elements … can be united
into a whole by some law.” But this somewhat more precise account of
what a set is does not make Cantor’s distinction any clearer. As we have
seen, the comprehension principle alone does not determine which collec-
tions are consistent and which are not. Indeed, it was precisely comprehen-
sion that led Russell to the contradictions. So again we are left only with
Cantor’s mentalistic intuitions about which sets exist. We have here just a
dispute of intuitions over what sets are like, while what was needed was
some definite criterion for set existence that did not lead to contradictions
in the theory. This is just what Zermelo and Russell would provide.

. Resolving the Paradoxes: Zermelo and Russell

In his  “Investigations into the Foundation of Set Theory I,”
Zermelo presented an axiom system for set theory that was “to exclude
all contradictions … [while] retain[ing] all that is valuable in this
theory.” While he does not spend nearly as much time directly consid-
ering the paradoxes as Russell does, Zermelo does state the aim of
his article as removing such difficulties from the theory. In opening the


Georg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers (, ),
trans. Philip E. B. Jourdain (New York, NY: Dover, ), p. .

Cantor, Foundations, , note .

Ernst Zermelo, “Investigations into the Foundations of Set Theory I,” in From Frege to Gödel, ed.
van Heijenoort, p. .
 Cantor, Russell, and Zermelo and the Set-Theoretic Paradoxes 
article, he first describes the important foundational work that set theory
makes possible, describing the theory as
that branch of mathematics whose task is to investigate mathematically
the fundamental notions of “number”, “order”, and “function”, taking
them in their pristine, simple form, and to develop thereby the logical
foundations of all arithmetic and analysis; thus it constitutes an indispen-
sable component of the science of mathematics.
He then immediately raises the issue of whether the theory can remain a
viable path for foundational research in light of the paradoxes:
At present, however, the very existence of this discipline seems to be threa-
tened by certain contradictions, or “antinomies”, that can be derived from
its principles – principles necessarily governing our thinking, it seems – and
to which no entirely satisfactory solution has yet been found.
Gregory Moore has argued that despite these remarks on the paradoxes,
Zermelo’s primary concern in his axiomatization is to give a precise
account of his proof of the well-ordering theorem. The proof had been
the subject of much controversy within the mathematical community
since its original appearance in . And in , before publishing
his axiomatization, Zermelo provided a second proof of the theorem,
which includes responses to his various critics. Moore describes
“Investigations into the Foundations of Set Theory” as a companion
paper aimed at making explicit exactly which principles the well-ordering
proof relied upon. Such emphasis has its place in correcting the histori-
cal record, since the more standard interpretation has overemphasized
the philosophical roots of set theory and the paradoxes. But a danger
arises in this revised interpretation as well. Too much emphasis on the
mathematical approach to set theory and with it the view that mathema-
tical set theorists were not greatly troubled by the paradoxes may lead to
the mistaken view that there were always two notions of a set, the mathe-
matical and the philosophical or logical, and that the mathematical


Ibid., p. .

Ibid.

The controversy arose particularly with regard to Zermelo’s use of the axiom of choice. I will
return to this topic in a later chapter. The original proof appears in Zermelo’s “Proof That Every
Set Can Be Well-Ordered,” pp. –.

Ernst Zermelo, “A New Proof of the Possibility of a Well-Ordering,” in From Frege to Gödel, ed.
van Heijenoort, pp. –.

Gregory H. Moore, “The Origins of Zermelo’s Axiomatization of Set Theory,” Journal of Philosophical
Logic  (), pp. –. Moore’s Zermelo’s Axiom of Choice: Its Origins, Development, and Influence
(New York, NY: Springer, ) is a book-length treatment of these issues.
 Quine, New Foundations, and the Philosophy of Set Theory
conception was never contradictory. While I want to concede that
mathematicians often reacted differently to the paradoxes than more philo-
sophically minded logicians, I also want to stress that all researchers in set
theory at the beginning of the twentieth century had to come to terms
with the paradoxes and that it was only in doing so that two conceptions
of set emerged. To the extent to which there was any agreed-upon notion
of a set prior to the paradoxes, it was roughly that a set is the extension of
a predicate or a concept. Both mathematical and philosophical logicians
had to clarify this notion of a set in light of the paradoxes. This is precisely
how Zermelo goes on to describe his axiomatization.
As we saw in the passage from Zermelo, he describes the existence of
contradictions in set theory as producing only a seeming threat to the
discipline, indicating that he does not think the paradoxes are an actual
threat. We should not take this to show that Zermelo never thought
there was a threat from the paradoxes. The threat is only apparent now
because there have been solutions to the paradoxes. We have seen already
Cantor’s attempt to resolve the paradoxes through his distinction
between consistent and inconsistent totalities. Zermelo himself had
already in his original  well-ordering proof appealed only to princi-
ples, which he believed yielded no contradictory results. Now his 
axiomatization can be added to this list. Still, he describes the current
situation in set theory as being that no completely satisfying solution to
the paradoxes has been found. We should not take this as indicating that
some better solution is yet to be found. Rather, Zermelo’s remark here
indicates his recognition of the limitations that the paradoxes have
imposed on set theory.
From Russell’s paradox especially, Zermelo explains, it seems no
longer possible for each logically definable notion to have a set or class as


The mathematical conception is typically identified with the iterative conception of set theory.
Quine himself, in The Roots of Reference (La Salle, IL: Open Court, , pp. –), recognized
this interpretation already in , observing that
A curiously myopic view of this matter has been manifesting itself of late [fn. omitted]. There
is hindsightful reaction, after two generations, to the paradoxes of set theory. The new view is
that even before the paradoxes it was not usual to suppose there was a set, or class, for every
membership condition. The view is defended by citing Cantor as having already entertained
certain restraints on the existence of classes before Burali-Forti published the first of the para-
doxes. Fraenkel has undercut this argument by claiming that Cantor already senses paradox
[fn. omitted]. What is myopic about the view … is that it looks back only to the first systema-
tic use in mathematics of the word “set” or “Menge”, as if this were uncaused.

Zermelo, “A New Proof of the Possibility of a Well-Ordering,” p. .
 Cantor, Russell, and Zermelo and the Set-Theoretic Paradoxes 
its extension, as had been long thought in the philosophical tradition.
Furthermore, Cantor’s original definition of a set as “a collection, gath-
ered into a whole of certain well-distinguished objects of our perception
or our thought” must also be restricted. What is unsatisfactory about
the proposed solutions, including his own, is that they do not preserve
these rather simple and intuitive notions of a set. These earlier accounts
of the notion of a set have not “been successfully replaced by one that
is just as simple and does not give rise to such reservations.”
Accepting this situation that our intuitions about sets have led us
astray, he concedes:
There is at this point nothing left for us to do but to proceed in the oppo-
site direction and, starting from set theory as it is historically given, to
seek out the principles required for establishing the foundations of this
mathematical discipline. In solving the problem we must, on the one
hand, restrict these principles sufficiently to exclude all contradictions
and, on the other, take them sufficiently wide to retain all that is valuable
in this theory.
Zermelo here acknowledges – continuing with our ongoing theme – that
after the paradoxes, developing set theory must be a largely pragmatic
endeavor. The aim in light of the paradoxes is to restrict the notion of
set sufficiently to exclude contradiction but to maintain enough of the
theory that it can continue its foundational purpose for all mathematics.
No solution to such paradoxes will be as intuitive as the notion of a set
as the extension of a concept, but this is the situation we must accept if
set theory is to go on at all. In this sense, no solution will be entirely
satisfactory, including Zermelo’s own; there will always be tradeoffs
about what to save from our initial intuitive account of sets.
Zermelo then presents his axiom system, attempting to show how he
can reduce Cantor’s set theory to a few definitions and seven axioms,
which appear independent of each other. To indicate that his axioms do
in fact preserve a reasonable amount of Cantor’s theory, he then develops
the theory of cardinal numbers and mentions that he will do the same
for the ordinal numbers in a later paper. He admits, however, that he
has not been able to give a rigorous proof of their consistency, “though
this is very essential.” Instead, he will merely indicate how his axioms


Cantor, Beiträge, p. . Georg Cantor, Contributions to the Founding of the Theory of Transfinite
Numbers (, ), trans. Philip E. B. Jourdain (New York, NY: Dover, ), p. .

Ibid., p. .

Ibid., p. .
 Quine, New Foundations, and the Philosophy of Set Theory
resolve the known paradoxes in hopes that this will provide a useful
beginning toward a consistency proof. Specifically, his separation
axiom, he explains, resolves the paradoxes. In contrast to the original
comprehension principle, sets can no longer be defined independently by
an axiom. Instead, they are always separated out as subsets of sets that
have already been given by other axioms. Therefore contradictory
notions such as “the set of all sets” or “the set of all ordinals” are
excluded from the theory. The defining condition of such subsets must
always be definite if the fundamental relations of the domain, meaning
the membership relation, by means of the axioms and the logical laws,
determine whether or not φ holds for a given value x. He says that in
addition to the set-theoretic paradoxes, this last restriction banishes the
semantic paradoxes such as the liar paradox and Richard’s paradox. It
should be noted that this idea of a propositional function being definite
would continue to be refined over the next twenty years or so until it
would come to mean “definable by means of first-order logic.”
Having explained separation, Zermelo next shows in his Theorem 
how Russell’s paradox now becomes the theorem that there is no set of
all sets. Every set M possesses at least one subset M that is not an ele-
ment of M. First, observe that “x is a member of x” is a definite property.
Now let M be a subset of M such that x is a member of M if and only
if x is not a member of itself. Then M is not a member of M. Suppose
M is a member of M. Then M is a member of M and M is not a
member of M by the construction of M. This is equivalent to saying
that M is a member of M if and only if M is not a member of M,
which is a contradiction. Hence M is not a member of M. So the
domain itself is not a set, and thus Zermelo’s theory eliminates Russell’s


Ibid., pp. –. Heinz-Dieter Ebbinghaus, in Ernst Zermelo: An Approach to His Life and Work
(New York, NY: Springer: , pp. –), criticizes Moore’s interpretation from the side of con-
sistency, observing that this was crucial to Hilbert’s program in the philosophy of mathematics.
Thus, much as I have been urging, Zermelo’s axiomatization is not a purely mathematical
undertaking.

Zermelo, “Investigations into the Foundations of Set Theory I,” pp. –. The liar paradox is
probably familiar to most readers and comes in a variety of forms. One of the simplest tells of the
person who asserts, “I am lying.” If this is a lie, then it is not. And if this is not a lie, then it is.
Richard’s paradox is more complicated but relies on the notion of definability and a diagonal
method of the sort found in Cantor’s proof that the set of real numbers is uncountable. If we let
X be the set of all definable real numbers, we have a countable set, since there are only countably
many definitions. If we order these definitions lexicographically, that is, in dictionary order, and
then apply the method of diagonalization to this ordering, we get a new real number that is not
in our set of definable real numbers. But have we not just defined this number by way of this
procedure?
 Cantor, Russell, and Zermelo and the Set-Theoretic Paradoxes 
paradox, turning it instead into a theorem. Let us now consider
Russell’s own resolution to the paradoxes.
Already in his Principles of Mathematics, Russell proposes solving the
paradoxes with his theory of types, a sort hierarchy of propositional
functions that would prevent such membership conditions as self-
membership and hence would exclude from the theory his own paradox,
among others. However, Russell’s proposal of types in the Principles is
tentative, and between  and , he explored a number of other
approaches to resolving the paradoxes before coming back to types in his
 “Mathematical Logic as Based on the Theory of Types.” He
begins here by remarking that this system of logic recommended itself to
him because of its ability to resolve certain paradoxes, in particular
Burali-Forti’s paradox of the largest ordinal number. “But the theory in
question seems not wholly dependent on this indirect recommendation,”
he argues, saying that “it has also, if I am not mistaken, a certain conso-
nance with common sense which makes it inherently credible. This,
however, is not a merit upon which much stress should be laid; for com-
mon sense is far more fallible than it likes to believe” [my emphasis].
We should note, here, how similar Russell’s approach is to Zermelo’s,
both in starting from the paradoxes and in motivating their proposed
solution by the success of resolving them. Neither Russell nor Zermelo
relies on some preconceived notion of set beyond just the mathematical
aims of the theory. Beyond this, there is no univocal conception of a set
that their respective renderings of the theory aim to capture. They each
recognize that it was our intuitions about sets that led the theory astray,
so these can no longer serve as the primary guide in further developing
the theory. For both Russell and Zermelo, what determines the success
of their axiomatizations is their ability to avoid the paradoxes while still
including enough of Cantor’s original theory to serve as a foundation for
mathematics. Again, like Zermelo, Russell takes Cantor’s attitude of
developing the theory in response to mathematical concerns and applies
it to the resolution of the paradoxes themselves.


Zermelo, “Investigations into the Foundations of Set Theory I,” p. .

Russell, Principles of Mathematics, pp. – and Appendix B.

This is particularly true of Russell’s  “On Some Difficulties in the Theory of Transfinite
Ordinals and Order Types,” in Essays in Analysis, ed. Lackey (London: George Allen and Unwin,
), pp. –. I will come back to this essay in discussing Quine’s Russellian background.

Bertrand Russell, “Mathematical Logic as Based on the Theory of Types,” in From Frege to Gödel,
ed. van Heijenoort, p. . This view, as we will see in Chapter , is strikingly similar to Quine’s
statement that in light of the paradoxes, “common sense is bankrupt.”
 Quine, New Foundations, and the Philosophy of Set Theory
Unlike Zermelo, Russell next gives a detailed account of the various
paradoxes, among them the liar paradox, the Burali-Forti paradox, and
his own. Russell’s detailed probing of the various paradoxes aims to
discover that they all share “a common characteristic.” The common
feature that Russell discerns is “self-reference, or reflexiveness.” So, for
example, when Epimenides the Cretan asserts that all Cretans are liars
and that all other statements made by them are lies, this assertion itself
must fall within the scope of Epimenides’ statement. Similarly, for all
classes x to belong to the class w if and only if class x is non-self-
membered, this condition must apply to itself. So, Russell explains, “[i]n
each contradiction something is said about all cases of some kind, and
from what is said a new case seems to be generated, which both is and is
not of the same kind as the cases of which all were concerned in what
said” [Russell’s italics]. He then proceeds through the paradoxes, show-
ing how this works in each case and concluding, “Thus all our contradic-
tions have in common the assumption of a totality such that, if it were
legitimate, it would at once be enlarged by new members defined in
terms of itself.” Thus Russell establishes the general principle – a
version of his vicious circle principle – that “[w]hatever involves all of a
collection must not be one of the collection” or, conversely, “[i]f,
provided a certain collection had a total, it would have members only
definable in terms of that total, then the said collection has no total.”
The theory of types will be a theory that excludes such collections,
making such phrases as “all propositions,” “all properties,” or “all classes”
meaningless. I should make clear at the outset of this discussion that
Russell’s theory is not the perhaps familiar modern type theory, now
known as the simple theory of types. Russell’s theory is what is called the
ramified theory of types. I will make the distinction clear, but let me
begin with Russell’s theory as he presents it in .
Russell begins by defining a type as the range of significance of a
propositional function, that is, the collection of arguments for which the
function has values or of which it is true or false. If a propositional func-
tion is applied to an argument outside of its range of significance, the
resulting proposition is not a proposition at all but rather is simply

Zermelo, “Investigations into the Foundations of Set Theory I,” pp, –.

Russell, “Mathematical Logic as Based on the Theory of Types,” p. .

Ibid., p. .

Ibid., p. .

Ibid. For a more detailed account of this principle and its motivations in Russell, see Peter
Hylton, “The Vicious Circle Principle,” in Propositions, Functions, and Analysis: Selected Essays on
Russell’s Philosophy (Oxford: Clarendon Press, ), pp. –.
 Cantor, Russell, and Zermelo and the Set-Theoretic Paradoxes 
nonsense. Whenever an apparent variable, that is, a bound variable,
occurs in a proposition, its range of values forms a type. The paradoxes
then are avoided by the vicious circle principle as implemented by the
theory of types: “No totality can contain members defined in terms of
itself” or, in terms of variables, “whatever contains an apparent variable
must not be a possible value of that variable.” Russell’s typing of the
objects of the universe restricts it so that any expression containing an
apparent variable must be of higher type than the possible values of that
variable. Thus the apparent variables in an expression determine the type
of the expression. He explains further that any proposition containing
an apparent variable is a “generalized proposition,” and any proposition
with no apparent variables is an “elementary proposition.” A generalized
proposition presupposes the other propositions from which it is obtained
by the process of generalization. Hence all generalized propositions pre-
suppose elementary propositions. In an elementary proposition, we dis-
tinguish one or more terms from one or more concepts, where the terms
are what we regard as the subject of the proposition and the concepts are
the predicates or relations asserted of the terms. Russell calls the terms of
an elementary proposition the “individuals,” and these form the lowest type.
In practice, however, he states that it is unnecessary to know which objects
belong to the lowest type or even to know, in a given context, whether the
lowest type are individuals. The lowest type in any context, then, may be
taken as the individuals, and all that is essential to them is the way in which
the other types are generated from them. All that ultimately matters are the
relative types of the objects in a given expression – a view that Russell
describes as “typical ambiguity” and a key insight to which we will see
W. V. Quine appealing in developing his own set theory.
With all of this in place, Russell’s theory then generates the type hier-
archy as follows. In generalizing the individuals in an elementary proposi-
tion, we generate a new proposition, and this will be a legitimate
procedure as long as no individuals are propositions, which is the case
because propositions are essentially complex whereas individuals lack all
complexity. So, for example, given the proposition “x is red,” where “x”
is taken as an individual variable, we may generalize on “x” to get the
new proposition “(∀ x)(x is red),” or “Every individual is red.” Since the
individuals cannot be propositions, generalizing on them in this
way does not lead to reflexive paradoxes. Russell calls the elementary


Russell, “Mathematical Logic as Based on the Theory of Types,” pp. –.

Ibid., pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
propositions along with the propositions containing only individual vari-
ables as apparent variables “first-order propositions,” and these propositions
form a new set of objects that can also be generalized so as to generate the
second type. Generalizing now on the variables of the second type, we gen-
erate the second-order propositions, which now also serve as quantifiable
variables so as to form the third type, and so on. Now we can see how the
typing removes the reflexive paradoxes. Epimenides’ statement now
becomes, for example, “All first-order propositions asserted by me are
false.” Since this is a proposition about first-order propositions, it is itself a
second-order proposition and so may be asserted truly. It does not assert
any true first-order proposition, and so does not yield a paradox as it pre-
viously did. In general, the (n + )th logical type consists of order n propo-
sitions, and these propositions contain as apparent variables propositions of
order no higher than n − . Since the types are mutually exclusive, it is
impossible to generate the reflexive paradoxes.
In practice, Russell explains that it is easier to work with a hierarchy of
functions than with one of propositions. Thus a function that takes only
individuals as arguments, thus always yielding a first-order proposition, is a
first-order function. A function involving only first-order functions or pro-
positions as apparent variables is a second-order function and so on.
Furthermore, he calls a function of one variable that is of order next above
its argument a “predicative function”; this is the same for propositions of
several variables. Hence the type of a function is determined by the type of
its values and the type and number of its arguments. Russell further
explains this hierarchy by denoting first-order functions of an individual
variable x as “φ!x.” Since no first-order function contains a function as an
apparent variable, hence forming a well-defined totality, we can generalize
on φ. Then any proposition with “φ” as apparent variable and no apparent
variable of higher type is a second-order proposition. If such a proposition
contains an individual variable x, then this proposition is not a predicative
function of x but it is a predicative function of φ, where φ is written
“f !(ψ!w
^ ).” So f is a predicative function, and its values form a well-defined
totality, so we can also generalize on f. This defines the third-order predica-
tive functions, which are functions that have third-order propositions as
values and second-order predicative functions as arguments and so on.
Let me step back a bit from Russell’s exposition here to make a bit
clearer how the ramified type theory works, in particular by clarifying the


Ibid., p. .

Ibid., p. .
 Cantor, Russell, and Zermelo and the Set-Theoretic Paradoxes 
relationship between types and orders. At this point, it will be useful to
introduce the perhaps more familiar modern simple theory of types, most
often attributed to F. P. Ramsey but discovered independently by Quine
as well in his  dissertation, “The Logic of Sequences.” Since Russell
always leaves his variables typically ambiguous, we will make some assump-
tions here not found in his exposition. The simple type hierarchy may be
described as follows. At the lowest type are the individuals; these we assign
type . Propositional functions that apply to individuals are of type . And
then functions that apply to these are of type  and so on. The types are
further distinguished according to how many arguments the functions
involved take. Since a propositional function of level  applies only to an
individual, self-reflexivity is ruled out here, since such a propositional func-
tion can never apply to itself without violating the type restrictions.
Clearly, such paradoxes as Russell’s are then blocked if we treat this theory
extensionally, that is, treat the propositional functions as classes and predi-
cation as membership. Then no class can be a member of itself.
Recall, however, that Russell’s vicious circle principle was implemen-
ted by way of restricting the range of the quantifiers, as encapsulated in
the form “whatever contains an apparent variable must not be a possible
value of that variable.” Here is where the theory of orders enters. The
simple type theory blocks the paradoxes of set theory by way of ruling
out self-membership; we just saw an example of this. The account is
slightly more complicated, but for similar reasons, Cantor’s paradox and
the Burali-Forti paradox are also ruled out. But as we also saw, Russell
initially shows instead how his theory blocks one of the semantic para-
doxes: the liar paradox (he later goes on to discuss how the set-theoretic
paradoxes are blocked by his ramified theory). The theory of orders is
aimed at these. In the liar paradox, we violate the vicious circle principle


F. P. Ramsey, The Foundations of Mathematics and Other Essays, Ed. R. B. Braithewaite
(New York: Humanities Press, ), pp. –.

More detailed accounts of the simple theory of types are abundant in the literature; see, for exam-
ple, A. A. Fraenkel, Y. Bar-Hillel, and A. Levy, Foundations of Set Theory, nd rev. edn
(Amsterdam: North-Holland, ), pp. –.

Russell did not distinguish between the set-theoretic and semantic paradoxes. He thought that a
successful resolution to the paradoxes would block both sorts. This is not an unreasonable
demand, given that his starting point was an intensional theory of propositional functions. I will
discuss this a bit more in Chapter . Ramsey is usually given credit for distinguishing the semantic
from the set-theoretic paradoxes, in the same paper in which he proposed his simple theory of
types. The simple theory, too, prevents the semantic paradoxes but by way of keeping metalin-
guistic claims concerning truth, definition, denotation, meaning, and the like distinct from the
object level claims that are assertible from within the system.
 Quine, New Foundations, and the Philosophy of Set Theory
by asserting a proposition, which would then itself fall into the range of
the apparent variable within that assertion. “All propositions asserted by
me are false” is itself a proposition and so would be a value of the appar-
ent variable used in talk of all propositions. The theory of orders prevents
such enlargement of the totality of propositions by prohibiting such use
of quantification. This is not the same as the simple theory of types.
To see this, consider the following two propositions:
() Napoleon is brave.
() Napoleon has all the qualities of a great general.
In the simple theory of types, the propositional functions “û is brave”
and “û has all the qualities of a great general” are of the same logical
type, since they both hold of individuals, assuming that people are
among the individuals. (Russell does not assume this.) In the ramified
theory, though, they both hold of individuals but are of different order,
since () would involve quantification over all the particular qualities,
such as bravery and leadership, that a great general has. Finally, we
should also distinguish Russell’s theory of orders from modern usage in
first-, second-, third-, and higher-order logics generally. The modern use
of these terms would allow quantification only over individuals, over
first-order functions, over second-order functions, and so on. Russell’s
theory allows for further possibilities. For example, functions of functions
of individuals are of second order, but so are functions such as “û has all
the qualities of a great general,” since these, too, involve quantification
over the totality of first-order functions.
We saw earlier with the example of Epimenides how Russell’s type
theory blocks at least some of the paradoxes, but now a new problem
arises. Recall that the success of his theory was to be judged on its capa-
city for providing a foundation to mathematics. As it stands, it generates
very little of mathematics. The typing restrictions we saw exclude such
expressions as “all propositions” or “all properties of x”; we can make
such assertions only when restricted to a particular order. But if a finite
number, for example, is defined in the usual way as a number that pos-
sesses all properties possessed by zero and by successors of all numbers


I have benefited in my exposition here from Alonzo Church’s “Comparison of Russell’s
Resolution of the Semantical Antinomies with That of Tarski,” Journal of Symbolic Logic :
(December ), pp. –, and from Bernard Linsky’s “The Notation in Principia
Mathematica,” https://plato.stanford.edu/entries/pm-notation/. Linsky’s exposition largely follows
Church’s but is less technical in nature and more attentive to the details of Russell’s actual
presentation.
 Cantor, Russell, and Zermelo and the Set-Theoretic Paradoxes 
possessing these properties, then type theory prohibits such a definition,
since it says we must confine our discussion to properties of a particular
order. If we confine this statement to all first-order properties of numbers,
then we cannot infer that this definition holds for all second-order proper-
ties. So, for example, we cannot prove that if m and n are finite numbers,
then m + n is a finite number, since by the previous definition, “m is a
finite number” is a second-order property of m. So from “m +  is finite,
and if m + n is finite, so is m + n + ,” we cannot conclude by induction
“m + n is a finite number,” leaving much of mathematics impossible.
Russell answers this difficulty by introducing his axiom of reducibility.
To regain mathematics, Russell needs a way of reducing the order of a
propositional function without affecting the truth or falsity of its values,
thus allowing him to simulate talk of all properties. To do so, he adopts
what he takes to be the commonsense reason for accepting classes. He
explains that, given any propositional function of any order, φx, we assume
the function to be equivalent, for all arguments x, to the statement “x
belongs to the class α.” This is a first-order statement, which makes no
mention of all functions of such-and-such a type, and its only practical
advantage over the original statement is its being of first order. Russell sees
no particular advantage in assuming outright the reality of classes (instead
defining classes contextually from his intensional propositional functions):
I believe the chief purpose which classes serve, the chief reason which
makes them linguistically convenient, is that they provide a method of
reducing the order of a propositional function. I shall, therefore, not
assume anything of what may seem to be involved in the common-sense
admission of classes, except this: that every propositional function is
equivalent, for all its values, to some predicative function.
Much as with Zermelo’s system and the basic system of types, the axiom
is introduced on purely pragmatic grounds. Russell, too, aims to preserve
as much of set theory as mathematics requires while still excluding the
paradoxes. Beyond this goal, there is no prior conception of a set for
which he aims. After all – as he was all too aware – the paradoxes showed
that neither common sense nor our intuitions about sets were reliable
guides in developing the theory.
Russell elaborates the new axiom, explaining that it is to hold for any
function, regardless of the type of its argument. So if we let φx be a func-
tion of any order of an argument x, which itself may be of any type,

Russell, “Mathematical Logic as Based on the Theory of Types,” p. .

Ibid.
 Quine, New Foundations, and the Philosophy of Set Theory
then by the axiom of reducibility, there is a predicative function φ!x that is
extensionally equivalent to it. This, Russell thinks, is all the usual assump-
tion of classes is meant to do and is as much of classes as he is willing to
assume. It is all he needs for mathematics. The axiom allows statements
about “all first-order functions” or “all predicative functions” to do the work
that statements about “all functions” previously did. “The essential point,”
he observes, “is that such results are obtained in all cases where only the
truth or falsehood of values of the functions concerned are relevant, as is
invariably the case in mathematics.” In other words, mathematics requires
only extensional equivalence. Yet there may be a worry that with the axiom,
the paradoxes return. Russell argues that this is not the case, since in the
case of the paradoxes, either something beyond the truth or falsity of the
values of the function is concerned or the expression occurs that remains
unmeaningful even after the axiom is applied, as he then goes on to show,
with regard to Cantor’s paradox and the Burali-Forti paradox.
Russell closes his paper much as Zermelo began:
The theory of types raises a number of difficult philosophical questions
concerning its interpretation. Such questions are, however, essentially
separable from the mathematical development of the theory and, like all
philosophical questions, introduce elements of uncertainty which do not
belong to the theory itself. It seemed better, therefore, to state the theory
without reference to philosophical questions, leaving these to be dealt
with independently.
It seems then that where set theory is concerned, the line between philo-
sophy and mathematics remains a difficult one to draw. But for Cantor,
Zermelo, and Russell, what must be preserved in set theory is its ability
to recapture ordinary mathematics, and they do so, guided by just this
pragmatic concern. We turn now to Quine and his set theory New
Foundations (NF), in which we will see this attitude toward set theory
developed more explicitly as a general approach to the subject.


Ibid., p. .

Ibid., pp. , –. For an excellent technical exposition of this point with regard to the het-
erological paradox and of the theory of types in general, see Church, “Comparison of Russell’s
Resolution of the Semantical Antinomies with That of Tarski,” pp. –, especially pp. –.

Russell, “Mathematical Logic as Based on the Theory of Types,” p. . It should be noted that
Moore attributes the discovery of Russell’s paradox to the important intertwining of mathematics
and philosophy. In “The Origins of Russell’s Paradox,” pp. –. Moore remarks, “Mathematics
and philosophy have interacted in many ways, but in the twentieth century mathematicians have
often reacted with suspicion when philosophers had something to say about mathematics. The
origins of Russell’s Paradox provide a case study of how traditional philosophical concerns led to
genuine mathematical progress.”
 CHAPTER 

New Foundations and the Beginnings of

Quine’s Philosophy of Set Theory

We now turn to consider the system of set theory put forward by W. V.

Quine in his  paper “New Foundations for Mathematical Logic.”
New Foundations (NF), as the system itself is known, emerged from
Quine’s thinking through the systems of both Bertrand Russell and Ernst
Zermelo, preserving something like the type restrictions of the one theory
along with the unrestricted variables of the other. Furthermore, Quine
adopts the preservation of set theory’s mathematical contributions – most
explicitly of Georg Cantor, Russell, and Zermelo – as the criterion of suc-
cess for his system. NF is especially notable – and controversial – in pro-
viding for the existence of such “big” sets as the set of ordinals, the set of
cardinals, and even the universal set. In doing so, it makes mathematical
sense out of Cantor’s absolute infinities, or supposedly inconsistent total-
ities, and shows that Russell was not entirely unjustified in his earlier com-
mitments to a class of all classes. In this chapter, I lay out the system and
then, as in Chapter , discuss how it avoids the paradoxes. This raises the
issue of the consistency of Quine’s system as well as consistencies for set
theories generally. I conclude with some remarks from a Quinean stand-
point on the mathematical and philosophical significance of consistency
proofs. I also begin to examine, from the perspective of Quine’s work in
logic and foundations, his more general philosophical approach to set
theory, which will be the primary focus of Chapters  and .

. From Russell and Zermelo to New Foundations

Quine begins by observing that Bertrand Russell and Alfred North
Whitehead’s Principia Mathematica provides substantial evidence that all
of mathematics is translatable into logic, although in light of Kurt

W. V. Quine, “New Foundations for Mathematical Logic,” The American Mathematical Monthly
: (), pp. –; reprinted in an expanded form in From a Logical Point of View,
pp. –.


 Quine, New Foundations, and the Philosophy of Set Theory
Gödel’s incompleteness theorem, no axiom system is sufficient for dedu-
cing all of mathematics. Still, the core of mathematics, as in Principia
Mathematica, serves as a reasonable standard for judging any system to
be sufficient for mathematics, so it is this standard that Quine adopts for
judging NF. Like Zermelo and Russell, Quine judges the success of any
given formulation of set theory according to whether or not the system
captures some reasonable amount of mathematics. As for logic, Quine
admits that the logic required for mathematics is a good deal stronger
than traditional logic:
It must be admitted that the logic which generates all this is a more
powerful engine than the one provided by Aristotle. The foundations of
the Principia are obscured by the notion of propositional function, but, if
we suppress these functions in favor of the classes and relations which
they parallel, we find a three-fold logic of propositions, classes, and rela-
tions. The primitive notion in terms of which these calculi are ultimately
expressed are not standard notions of traditional logic; still they are of a
kind which one would not hesitate to classify as logical.
So we have Quine clarifying exactly what he will count as a logical foun-
dation for mathematics: logic inclusive of set theory. But in accepting
such a position, he knowingly gives up much of the epistemological aims
that the reduction of mathematics to logic was supposed to accomplish.
Quine’s concern is not to somehow rest mathematics on a firmer founda-
tion than mathematics itself. Rather, this foundation will be internal to
mathematics itself. However, for Quine, set theory, as we will see later,
remains foundational in that it unifies diverse bodies of mathematics into
the single framework of logic and set theory. This theme is important
for understanding Quine’s philosophy of set theory as a whole, which we
will deal with more thoroughly in the following chapters. For now, we
merely observe that these departures from traditional philosophical aims
are present in some of his earliest work in foundations. However, this is
not the only point of interest here in beginning to flesh out Quine’s phi-
losophical stance. We also see him in this passage quickly dismissing
Russell’s basic notion of a propositional function on grounds of its

Ibid., p. .

This point is also interesting in light of later disputes over the logical status of higher-order logic.
We see here that Quine was already in a position at this early stage to draw a distinction between
first- and higher-order logic on grounds of their respective claims to logicality. In a way, what we
see here is that the label “logic” was never of great importance to Quine. What was important was
making clear what it was we were willing to count as logic and being honest about what sorts of
assumptions we are building into attributing the label “logic.” Among such assumptions, as in the
case here, is the mathematical strength of such theories.
 New Foundations & Beginnings of Quine’s Philosophy 
obscurity and instead accepting classes outright. The central Quinean
themes of clarity and simplicity are present already at this early stage,
and in recognizing the obscurity of the notion of a propositional func-
tion, we can see the beginnings of Quine’s attack on philosophical
accounts of meaning and so also on the analytic/synthetic distinction.
We see here core concerns of Quine’s philosophy emerging from his logi-
cal work. Let us turn now to the system itself.
The system of NF contains a much smaller number of basic notions
than that of Principia Mathematica, employing just membership, “∈”
(epsilon); alternative denial, “|” (not both); universal quantification, “(∀x)”;
and an infinite number of variables such as “x,” “y,” “z,” “x′,” “y′,” “z′,”
“x″,” and so on. The variables in NF, unlike those of Principia
Mathematica, range over all objects whatsoever; there is no stratification
of the universe into different levels or types. From these basic notions,
Quine then defines the usual logical connectives of negation, conjunction,
disjunction, conditional, biconditional, and the existential quantifier, along
with the various set-theoretic relations such as subset, “⊆”; union, “∪”; and
intersection, “∩.” In remarking that the formula “(x ∈ y),” or “x is a mem-
ber of y,” makes sense only where y is a class, Quine further simplifies the
system by adopting the convention that where y is an individual, we may
interpret such formulas as “x is the individual y.” As such, every individual
x is just equal to its unit class, or singleton, {x}, but Quine adds in a foot-
note that “this is harmless.” Again, we see the broadly pragmatic spirit that
has come to characterize Quine’s philosophy emerging in his logical work.
Where he can simplify the system, unlike – and in reaction to – Russell, he
does so without regret.
Next, Quine introduces the basic rules and axioms, or initial theo-
rems, of NF, among them truth-functional and quantificational axioms
and rules of inference. He also proposes the following two set-theoretic
principles:
P. (∀x)(∀y)(x ⊆ y ⊃ ( y ⊆ x ⊃ x = y)),
the extensionality principle that a class is determined by its members, and
R. If “x” does not occur in φ, then “(∃x)(∀y)(( y ∈ x) ≡ φ)” is a
theorem.


Quine, “New Foundations for Mathematical Logic,” p. . This convention is indeed harmless, as
shown by Scott, but it also results in the surprisingly very different system of NFU, which was
proved consistent in Peano arithmetic by Jensen.
 Quine, New Foundations, and the Philosophy of Set Theory
In short, given any condition on y, there is a class x with members y
such that this condition holds for the y’s. Quine recognizes that this is,
of course, the inconsistent comprehension principle, which yields
Russell’s paradox when we take φ as the condition “( y ∉ y).” As we saw
in Chapter , one way of resolving the paradoxes was to adopt types so
that the universe is stratified into levels and the variables are restricted to
ranging over objects of a particular level. Thus (α ∈ β) is a formula only
if the values of β are of type n +  and the values of α are of type n.
In all other cases, (α ∈ β) is neither true nor false but meaningless.
Quine, however, focuses his attention on an observation that we saw
Russell himself make in the previous chapter:
In all contexts the types appropriate to the several variables are actually
left unspecified; the context remains systematically ambiguous, in the
sense that the types of its variables may be construed in any fashion con-
formable to the requirement that “∈” connect variables only of consecu-
tively ascending types. An expression which would be a formula under
our original scheme will hence be rejected as meaningless by the theory of
types only if there is no way whatever of so assigning types to the variables
as to conform to this requirement on “∈.” Thus a formula in our original
sense of the term will survive the theory of types if it is possible to put
numerals for the variables in such a way that “∈” comes to occur only
in contexts of the form “n ∈ n + .”
While Russell had already noticed this feature of types, Quine follows it
through to its full conclusion. It is inessential to resolving the paradoxes
that the universe actually be stratified into levels. What the typing restric-
tion does is to provide a syntactic test for which formulas actually deter-
mine classes; these are the so-called stratified formulas. NF, in this sense, as
Quine later remarked, emerged from the theory of types. Here, we get
early hints of the philosophical attitude he adopts toward set theory. His
philosophical vision is served if he can somehow simplify the theory and,
furthermore, avoid some of the metaphysical assumptions about set theory
that Russell’s hierarchy of types has often inspired.
To help motivate this move, Quine notes some unnatural conse-
quences of Russell’s type theory. Although we saw Russell clinging despe-
rately to a universal class, his resolution of the paradoxes in type theory
ultimately forced him to give up this idea in favor of a series of what

Ibid., p. .

Ibid., p. .

W. V. Quine, “The Inception of New Foundations,” in Selected Logic Papers, enlarged edn
(Cambridge, MA: Harvard University Press, ), p. .
 New Foundations & Beginnings of Quine’s Philosophy 
Quine describes as quasi-universal classes, a series of classes of all objects
of the type directly below a given type. So there is a class of all indivi-
duals, a class of all classes of individuals, and so on, but there is no single
class V that contains all objects regardless of type. Similarly, there is no
single null class Λ but, again, one for each type. Type theory has the
further odd consequence that classes no longer have absolute comple-
ments. A class –x does not contain all nonmembers of x but rather only
those nonmembers of x at the next lower type from x. So we see that the
Boolean class algebra no longer applies to classes in general but only to
classes within particular types. Furthermore, these classes are then redu-
plicated within each type, and the same holds also for the usual calculus
of relations. Finally, Quine notes that even arithmetic is reduplicated in
each type so that the numbers fail to be unique, there being for each
type a number , , , , and so on. He concludes, “Not only are all
these cleavages and reduplications intuitively repugnant, but they call
continually for more or less elaborate technical maneuvers by way of
restoring severed connections.” By taking Russell’s insight of typical
ambiguity seriously, Quine resolves the paradoxes while avoiding these
unhappy consequences: “Whereas the theory of types avoids the contra-
dictions by excluding the unstratified formulas from the language all
together, we might gain the same end by continuing to countenance
unstratified formulas but simply limiting R explicitly to stratified for-
mulas.” The hierarchy of types is then removed so that variables stay
unrestricted in range and the logical language allows for all formulas in
the original untyped sense. The notion of a stratified formula remains a
part of the system only in the following revised version of R:
R′. If φ is stratified and does not contain “x,” then “(∃x)( y)(( y ∈ x) ≡
φ)” is a theorem.
Here, we also get a glimpse of a further reason for rejecting the theory
of types that Quine was to emphasize in his later account of how he
came upon NF. There, he explained that he disliked the arbitrary gram-
matical restrictions of type theory. As was mentioned previously, unstra-
tified formulas, according to type theory, were neither true nor false but
meaningless, so “[s]eemingly intelligible combinations of signs were
banned as ungrammatical and meaningless.” The theory of types was,


Ibid., pp. –.

Ibid., p. .

Quine, “Inception,” p. .
 Quine, New Foundations, and the Philosophy of Set Theory
for Quine, too drastic in its resolution of the paradoxes in banning all
unstratified formulas, especially since many of them seem perfectly intel-
ligible. Much of his work up to NF was an attempt to legitimize these
restrictions so that the banning of unstratified formulas would not be
wholesale and without reason. In his first attempt in , he tried to
legitimize the restrictions by finding some more basic array of notions
that would allow him to provide contextual definitions of classes and the
membership relation. “I hoped,” he explained, “to devise contextual defini-
tions that would generate just the formulas that fit the theory of types,
while leaving other formulas meaningless in the straightforward sense of not
being accounted for by the definitions.” The attempt failed, however, as the
set-theoretic paradoxes simply shifted to become semantic paradoxes.
At this point, the crucial influence on Quine shifted from Russell to
Zermelo. From Zermelo, Quine gained the insight “that a meaningful
open sentence may or may not determine a class, and that it can be left
to the axioms to settle which ones do.” So while Russell banned such
open sentences as “x is a member of itself” on the grounds of their sup-
posed meaninglessness, Quine took the contrary position, affirming their
meaningfulness but leaving it open that they may not determine a class,
depending on which classes the axioms yield. His first step in the direc-
tion of NF was the  “Set-theoretic Foundations for Logic,” in which
he presents a version of Zermelo’s system but with a modified version of
the separation axiom. Zermelo states his original version of separation as
ð∃ xÞð∀yÞð y ∈ x ↔ ðx ∈ z ∧ FyÞÞ:
Quine’s idea was to replace “x ∈ z” with “x ⊆ z” so as to obtain
ð∃ xÞð∀yÞð y ∈ x ↔ ðx ⊆ z ∧ FyÞÞ:
While Zermelo’s system required several other set-theoretic axioms to be
adequate for the derivation of mathematics, Quine’s modified system,
which he called “Γ,” included only the modified separation axiom and
the extensionality axiom. In line with the motivations for NF, his aim in
“Set-theoretic Foundations” was “the presentation of a system Γ which
resembles [Zermelo’s] system but is more economical.” Like the sys-
tems of Zermelo and Russell, however, to derive a substantial amount of
mathematics (namely, all of what could be derived in the system of

Ibid.

Ibid., pp. –.

Ibid., p. .

W. V. Quine, “Set-Theoretic Foundations for Logic,” in Selected Logic Papers, p. .
 New Foundations & Beginnings of Quine’s Philosophy 
Principia Mathematica), the system Γ still required postulating the addi-
tional axioms of infinity and choice.
Indeed, this lack of economy was the primary drawback Quine saw in
Zermelo’s system, since it otherwise avoided the difficulties of types in
that it lacked both arbitrary grammatical restrictions and the reduplica-
tion of objects at each level of the hierarchy: “Zermelo’s system itself
was free of both drawbacks, but in its multiplicity of axioms it seemed
inelegant, artificial, and ad hoc.” But both Zermelo’s system and the
modified version Γ had another drawback in removing type theory’s
reduplication of objects. While types had quasi-universal classes,
Zermelo’s system banned this class altogether; therefore, absolute com-
plements were again impossible, yielding instead only complements with
regard to an already given class. In general, what both systems lacked
were big classes. Here, we see another aspect of Quine’s philosophy of
set theory, which will emerge more fully in Chapter . Quine approaches
set theory as an explorative project much like the rest of science. The
lack of big classes is not necessarily a drawback for reconstructing mathe-
matics within set theory. Most ordinary mathematics goes on with com-
paratively small sets, but such a lack becomes all the more apparent as
set theory itself has emerged as a subject of mathematical investigation.
In exploring the mathematical behavior of the set-theoretic universe, we
should not limit our explorations to only some sets. We should want to
know what all sets are like, how they are similar and where they diverge,
and what sorts of features cause such divergences. If we want to know
what the big sets are like, neither the system of Russell nor that of
Zermelo can give us an answer.
What Quine wanted was a system that would be free of all the draw-
backs of both systems “but would be like the theory of types in having a
single comprehension principle of class existence, and would admit big


In later chapters, we will see more of how Quine revised this view of Zermelo’s set theory in light
of the connections he was able to draw between it and type theory. Drawing out these sorts of
connections is central to Quine’s philosophy of set theory and to his philosophy in general. He
gestures at this move in “The Inception of New Foundations,” pp. –. Contrast this with his
earlier statement in “New Foundations for Mathematical Logic” (p. , fn.) about the awkward-
ness of Zermelo’s system, which he notably removed from the paper in the form published in
From a Logical Point of View.

Quine, “The Inception of New Foundations,” p. .

For this history, see Kanamori’s “Development,” pp. –.

This is perhaps very much in the spirit of much current research in set theory that is concerned
with large cardinal axioms. We want to know what the universe is like beyond the sets given by
ZFC. Yet NF remains largely neglected in its possible insights into the universe.
 Quine, New Foundations, and the Philosophy of Set Theory
classes without restriction to type.” Combining the insights of Russell’s
typical ambiguity with Zermelo’s view that the axioms could determine
which sets exist gave Quine precisely this, as codified in his system NF.
In light of Zermelo, “I was able to look to types as a restriction specifi-
cally upon classes and not upon language,” Quine observed. “The pur-
pose of the theory of types was to bar the paradoxes, and this could be
done by using it only to say which open sentences are to be taken to
determine classes.” The structure of the set-theoretic universe, then,
was determined by the stratification condition limited only to the com-
prehension principle R′. The universe itself was not stratified into a
hierarchy of classes, as Russell thought. Now, with the variables recon-
strued as general, that is, as ranging over all objects with no restriction of
type, Quine “found new strength accruing at every turn, apparently with
impunity.” NF successfully removed Russell’s arbitrary grammatical
restrictions while not adding the reduplication of objects back in.
Furthermore, he regained the universal class and absolute complements
as well as other such big classes as the class of cardinals and the class of
ordinals. Finally, he obtained an infinite class without the arbitrary pos-
tulation of an axiom of infinity in that the universal class existed and
contained within in it each natural number such that for all such num-
bers m and n, m ≠ n.
Quine’s aim in presenting NF was not to do away with other set the-
ories, to have somehow presented the single “right” conception of a set.
Rather his aim was, and would remain, to further set-theoretic research
generally by looking at sets from a different perspective, which preserved
certain fairly intuitive features of sets that the systems of Russell and
Zermelo did not. From Quine’s perspective, none of these systems dis-
covered anything like the essence of sets so as to discover the one and
only correct set theory. Rather, just like Russell’s and Zermelo’s, Quine’s
primary purpose was eliminating the paradoxes while preserving enough
of Cantor’s original theory for it to be mathematically interesting. Any set
theory that was adequate to these aims was one worthy of mathematical –
or perhaps we might even say more broadly – scientific investigation. As
we will see in Chapter , Quine’s philosophy of set theory can be encapsu-
lated in the notion of explication as opposed to conceptual analysis.


Quine, “The Inception of New Foundations,” p. .

Ibid.

Ibid.

However, it would not be until  that Ernst Specker showed outright that the axiom of infi-
nity was provable in NF.
 New Foundations & Beginnings of Quine’s Philosophy 
. “Contradictions Really Scare Me”: New Foundations and the
Paradoxes
But did NF in fact avoid the paradoxes? Quine remarks in conclusion
that he has no proof that such a system is consistent, but just as Zermelo
and Russell reasoned about their respective systems, Quine can see no
way of deriving a paradox within it. He himself makes this connection:
“The lack of a consistency proof is no special ground for misgivings, for
there is likewise none for the systematization involving the theory of
types.” Furthermore, his system removes the various unnatural conse-
quences of types. Both the universal class V and the null class Λ are
unique, while the complement of x, –x, is the class of everything not in
x. Thus the Boolean class algebra is restored and similarly for the calculus
of relations. The reduplications of arithmetic are also removed so that
the numbers are unique and their laws become generally applicable as a
single calculus, thus eliminating whatever complicated technical devices
type theory required for restoring such severed connections.
In concluding his paper, Quine remarks that NF differs from the origi-
nal inconsistent theory in preventing the existence of classes defined by
unstratified conditions. He adds in a footnote that the systems of his ear-
lier “Set-theoretic Foundations for Logic” and Zermelo’s offer other
approaches for resolving the paradoxes. “But these methods entail most of
the awkward limitations which are entailed by the theory of types,” he
argues. “The present method of avoiding the contradictions, if it indeed
avoids them, would seem to be the least restrictive method yet
suggested.” So we see again that Quine’s guideline for developing set
theory is much like Zermelo’s and Russell’s: to preserve as much as the
original theory unrestricted theory as possible. In this sense, NF does seem
an advance over Zermelo’s system and theory of types. While both of these
versions of set theory are sufficient for reconstructing mathematics, they
lack certain sets, the so-called big sets in particular. As we have seen, these
other theories include a variety of inelegancies resulting from the way in
which they restrict the notion of set so as to avoid the paradoxes.
So far, we have had only Quine’s speculations that NF is free from
contradiction. It does seem fairly obvious that Russell’s paradox will not


The quote in the section title comes from my friend, Jon Snodgrass; see his “History with
History,” from the album Drag the River (Last Chance Records, ).

Quine, “New Foundations for Mathematical Logic,” p. .

Ibid.

Ibid., p. .
 Quine, New Foundations, and the Philosophy of Set Theory
be a source of contradiction in the system, given that there is no set with
the membership condition y ∉ y. But how exactly the theory avoids
the other paradoxes, such as Cantor and Burali-Forti, is less clear.
Furthermore, as we saw in Chapter , Cantor had arrived at an impor-
tant result of set theory – the theorem bearing his name – that the power
set of a set always has greater cardinality than the set itself. It seems that
we should want to preserve such a result if we are to preserve Cantor’s
set theory. I now examine how in fact the known paradoxes are blocked
in NF and also how the theory preserves an analog of Cantor’s theorem.
Recall from Chapter  that Cantor’s paradox results from the claim
that there exists a universal set and Cantor’s result that for each set x, the
power set of x has greater cardinality than the set x itself. Quine’s NF
certainly yields a universal set V in its stratified existence claim that
(∃x)(∀y)( y ∈ x ≡ y = y). So it seems that such a set should have the
greatest cardinality, given that it contains all sets. But then by Cantor’s
theorem, the cardinality of the power set of V would be greater than V
itself, landing us back in the contradiction that worried Russell so much.
How then does NF block Cantor’s paradox? In the first general result
about NF, Quine himself showed how his system both blocks this para-
dox and preserves an analog of Cantor’s theorem. Although it may seem
tedious, it is instructive to follow the details of this proof to see exactly
how stratified comprehension works in preventing Cantor’s paradox.
Quine states Cantor’s theorem as “the converse domain of any one-
many relation has a subset which does not belong to the domain.”
Formally,
ð∀vÞ½ð∀yÞð∀zÞð∀wÞððz; yÞ ∈ v∧ðw; yÞ ∈ v ⊃ z = wÞ⊃
(.)
ð∃ xÞð∀yÞð y ∈ x ⊃Þð∃ zÞðz; yÞ ∈ vÞ∧ð∃ yÞðx; yÞ ∈ vÞ:
Here, he renders functions in terms of two-place relations, where a rela-
tion Rxy is just a set v with the ordered pair (x, y) as a member. We can
then read it as “There is a set x, a subset of the converse domain of R,
such that for any object y, if it is in x, then it is borne R by something,


Of course, if some other contradiction should be found in the system, then Russell’s paradox
along with anything else will be derivable as well.

Claims using identity are stratified when the objects on each side of the relation are of the same
type in a stratified formula. Quine takes identity as defined in terms of the identity of indiscern-
ibles: x = y = df (∀z)(x ∈ z ⊃ y ∈ z). So we could also replace occurrences of identity with this
formula, checking the resulting formula for stratification.

W. V. Quine, “On Cantor’s Theorem,” The Journal of Symbolic Logic : (September ),
pp. –.
 New Foundations & Beginnings of Quine’s Philosophy 
but the set of such y’s bears R to nothing, i.e., is not in the domain
of R.” Hence we have that a set always has more subsets than members.
To prove (.), assume that v is a one-many relation, where a one-
many relation is a relation such that any two members of the domain of
the relation v bearing this relation to the same member in the range
must be the same object, that is,
ð∀yÞð∀zÞð∀wÞððz; yÞ ∈ v ∧ðw; yÞ ∈ v ⊃ z = wÞ: (.)
In other words, such a relation is a function, since for any object in the
domain, or input, there is a unique object in the range, or output, to which
it bears the relation. By unrestricted comprehension, let x be the set
ð∀yÞð y ∈ x ↔ ð∃ zÞððz;yÞ ∈ v ∧ y ∉ zÞ: (.)
So if y ∈ x, then y ∉ z, and so z ≠ x. So from (.) and truth-functional
logic,
ð∀yÞð y ∈ x ⊃ð∃ zÞððz;yÞ ∈ v ∧ z ≠ xÞ: (.)
But then by (.) and truth-functional logic,
ðz; yÞ ∈ v ∧ z ≠ x ⊃ðx; yÞ ∉ v: (.)
Therefore by (.) and (.) and truth-functional logic,
ð∀yÞð y ∈ x ⊃ðx;yÞ ∉ vÞ: (.)
Now, from (.) again,
ð∀yÞðx; yÞ ∈ v ∧ y ∉ x ⊃ y ∈ xÞ; (.)
which, by truth-functional logic, is equivalent to
ð∀yÞðx; yÞ ∈ v ⊃ y ∈ xÞ: (.)
so from (.)
ð∀yÞððx; yÞ ∈ v ⊃ðx; yÞ ∉ vÞ; (.)
which is equivalent to
ð∃ yÞðx; yÞ ∈ v: (.)
From (.),
ð∀yÞðy ∈ x ⊃ ð∃ zÞððz; yÞ ∈ vÞ: (.)
Thus from (.) and the unrestricted comprehension principle, (.)
follows, establishing Cantor’s theorem.
 Quine, New Foundations, and the Philosophy of Set Theory
As we have seen, unrestricted comprehension led to the set-theoretic
paradoxes. Indeed, from this principle, we have seen that we can prove
the existence of a universal set, so Cantor’s paradox would quickly fol-
low. In Zermelo’s system, we can also prove Cantor’s theorem, but his
restriction on comprehension excludes the existence of a universal set, so
there is no reason to worry about the reemergence of the paradox there.
But NF, like set theory with unrestricted comprehension, allows for a
universal set, so how does it prevent Cantor’s paradox?
The proof of Cantor’s theorem follows from comprehension, so it will
likewise follow from comprehension in NF if the existence condition is
stratified. However, it is not stratified. The instance of comprehension
used earlier was
ð∃ xÞð∀yÞðy ∈ x ≡ ð∃ zÞððz; yÞ ∈ v ∧ y ∉ z: (.)
But using the Wiener-Kuratowski definition of ordered pair and check-
ing for stratification, we get
ð∃ xÞð∀yÞðy ∈ x ≡ ð∃ zn Þð{{zn }n+1 ; {zn ; yn }n+1 }n+2 ∈ vn+3 ∧ yn ∉ zn Þ;
(.′)
demonstrating that this is an unstratified condition, since y and z must
have consecutive subscripts n and n + . Therefore no such set exists
in NF, and Cantor’s paradox is blocked. But it also follows now that
Cantor’s theorem in its stated form is false, and this may appear to
weigh heavily against NF as a live option for formalizing set theory,
since it now seems to prevent one of Cantor’s most important results
about set theory. It is central to investigating infinite sets, and losing it
would remove much of what made Cantor’s theory interesting in the
first place.
Quine, however, observes that NF does have an analog to Cantor’s
theorem. Indeed, it is the same version of the theorem that we can prove
for type theory, since there, too, the stated version fails in its violation of
the typing restrictions. In type theory, relations can be rendered as sets
of ordered pairs in which the members of the pairs must be of the same
type. Mathematics, however, frequently requires pairs with members of
different types. Without some way around this, mathematics would be
all but impossible in type theory. The solution was to use the singleton
operation as a way of raising types. So, for example, the pair (x, y),
which is excluded by typing restrictions, can be rendered in type theory
in the following way. Again using the Wiener-Kuratowski analysis of
the ordered pair, we get {{x}, {x, y}}. We then apply the singleton
 New Foundations & Beginnings of Quine’s Philosophy 
operation to appropriately raise the type so as to get {{{x}}, {{x}, y}},
which is then the stratified ordered pair ({x}, y), a perfectly acceptable
statement of type theory.
Similarly, we can now replace the original statement of Cantor’s theo-
rem () with
ð∀vÞ½ð∀yÞð∀zÞð∀wÞððz; {y}Þ ∈ v ∧ðw; {y}Þ ∈ v ⊃ z = wÞ
(.′)
⊃ ð∃ xÞðð∀yÞðy ∈ x ⊃Þð∃ zÞðz; {y}Þ ∈ vÞ∧ð∃ yÞðx; {y}Þ ∈ vÞ;
which is derivable in NF, since the condition (∃zn+)((zn+, {yn}n+)n+ ∈
vn+ ∧ yn ∉ zn+) is stratified, hence allowing for the existence of the
required set. However, where () states that the subsets of a set cannot
be correlated one-to-one with the set’s members, that is, there are
more subsets than members, (′) states that a given set has more sub-
sets than singleton subsets, which has the further consequence that that
in general there is no function correlating one-to-one members with
their singletons. While this last result may seem especially odd, there
are many sets for which such a correlation exists, such as the so-called
Cantorian sets, to which we will return in later chapters. To conclude,
what Quine shows is that for NF, the original version of Cantor’s the-
orem is false, although an analog of it can be proved in the form that
the subsets of a set outnumber the singleton subsets, and that there is
no correlation in general between the members of set and their single-
ton sets.
The next major investigation into the consistency of NF came in 
with Rosser’s “On the Consistency of Quine’s New Foundations for
Mathematical Logic.” Here, Rosser presents a stronger system, Q, and
shows that none of the usual methods for producing contradictions are
possible in this system. Q also yields all of Quine’s system NF. Therefore
if Q is free from contradiction, so is NF.
Rosser begins by remarking that Quine’s result that Cantor’s paradox
is not derivable in NF holds similarly for Rosser’s own system Q.
However, he is able to generalize Quine’s findings to show how this
result shows the various other set-theoretic paradoxes to be similarly
underivable for both Q and NF. Rosser observes that the central point of
Quine’s proof is his showing that there is no function that takes every set
to its singleton. That is, there is no set of ordered pairs (x, y) such that x
is equal to y where y is the singleton of x, {(x, y): x = {y}}. Indeed, the

J. Barkley Rosser, “On the Consistency of Quine’s New Foundations for Mathematical Logic,”
The Journal of Symbolic Logic : (March ), pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
existence of such a set would also yield both the Russell and Burali-Forti
paradoxes. For example, consider the following:
ð∀RÞð∃ αÞð∀xÞðx ∈ α ≡ ð∃ yÞðx ∉ y ∧{x}RyÞÞ:
Now, instantiating R by {(x, y): x = {y}} yields
ð∃ αÞð∀xÞðx ∈ α ≡ ð∃ yÞðx ∉ y ∧{x} = {y}Þ;
which implies
ð∃ αÞð∀xÞðx ∈ α ≡ x ∉ xÞ;
the Russell class. In fact, Rosser concludes that for any unstratified condi-
tion φ, we can find a stratified condition ψ with free occurrences of a
relation R such that replacing R with {(x, y): x = {y}} makes ψ equivalent
to φ. Since relations are defined as classes of ordered pairs, which in turn
are just classes of classes, they depend on stratified comprehension for
their existence. Hence attempts to prove the existence of unstratified rela-
tions so as to derive the Russell, Cantor, or Burali-Forti paradox fail in
NF. So Rosser improves upon Quine’s result by showing that this gen-
eral feature of NF, that is, that in general there is no function taking sets
to their singletons, blocks the usual ways of generating the set-theoretic
paradoxes in NF, thus furthering the plausibility of NF’s consistency.
While a similar argument holds against the Burali-Forti paradox, it is
helpful to consider in some detail, as with the Cantor paradox, how
exactly it plays out in NF. Unfortunately, it requires a bit more technical
machinery than did the discussion of the previous two paradoxes. First,
recall that the Burali-Forti paradox is the paradox of the greatest ordinal.
Let ON be the set of ordinals that is naturally well ordered by ≤, and let
Ω be the order-type of ≤ restricted to ON. Since Ω is the type of ≤ on
the ordinals, Ω is greater than all the ordinals in ON, but Ω is also
an ordinal, so it must be a member of ON. Hence Ω is both greater
than all the ordinals in ON and one of the ordinals in ON.
To begin our discussion of how the Burali-Forti paradox plays out in
NF, we must first understand which set-theoretic objects the ordinals are
in NF. The cardinals of NF must be Russell–Whitehead cardinals, that
is, equivalence classes under equinumerosity, so, for example, the

Ibid., p. .

Rosser does this in his later Logic for Mathematicians (New York, NY: McGraw-Hill Book
Company, ), pp. –. Here, I primarily follow Randall Holmes’s presentation in Elementary
Set Theory with a Universal Set, Cahiers du Centre de Logique, vol.  (Louvain-la-Neuve:
Academia, ), pp. –. Holmes is somewhat more accessible than Rosser.
 New Foundations & Beginnings of Quine’s Philosophy 
cardinal number of a set x is the set of all objects the same size as x.
Similarly, the ordinals of NF must be Russell–Whitehead ordinals.
So for a well-ordering (X, R), its ordinal number is the set of all well-
orderings order isomorphic to it. Recall that every ordinal α has a suc-
cessor α +  such that α +  > α and they are naturally well ordered by
the relation ≤. For α an ordinal, we define an (initial) segment deter-
mined by α, written “seg≤{α}”, as the set {β ∈ NO: β < α}. In a strati-
fied formula, seg≤{α} is one type higher than α. As with the cardinal
numbers, type-raising operations will be important if we are to maintain
stratification restrictions, so we introduce the operation T. If α is the
order-type of (X, R), then T{α} is the order-type of (P{X}, RUSC(R)),
where P{X} is the set of singleton subsets of X and RUSC(R) is the set
of relational singleton (or unit) subsets of R, that is, the set {({x}, {y}):
xRy}. Iterations of the operation T are written “T n” for n a natural num-
ber. A segment seg≤{α}, then, is a member of the uniquely determined
ordinal T {α}, which is two types higher than α in a stratified formula.
Now we sketch a version of the Burali-Forti paradox. Consider the fol-
lowing inductive argument that for all ordinals α, T {α} = α. Let β be
the smallest ordinal such that T {β} ≠ β. For each ordinal α < β, then,
T {α} < T {β}. Furthermore, we see that every ordinal less than T {β}
must be T {γ} for some γ < β. But then it follows that T {β} is the smal-
lest ordinal greater than all the ordinals T {α} for α < β. That is, T {β}
is β itself. Hence it follows that there is no such β such that β ≠ T {β}.
So for all ordinals α, T {α} = α. Next, we apply this result to the well-
ordering ≤ itself. We let Ω be the ordinal containing ≤. By this argu-
ment, T {Ω} = Ω. Since Ω is the ordinal of ≤, the ordinal of ≤
restricted to seg≤{Ω} is also Ω. Here arises the contradiction. Since the
well-ordering ≤ is a strict continuation of seg≤{Ω}, ≤ must also include
additional ordinals Ω + , Ω + , and so on greater than the purported
greatest ordinal Ω.
Fortunately, Quine’s NF does not allow for this argument to go
through. Since T {α} and α are of different types in a stratified formula,
T {α} = α is not stratified and hence does not determine a set. Since we
can argue by transfinite induction only with regard to sets, the argument
then fails because T {α} = α and is an unstratified condition. Now we
again see Rosser’s general point about the singleton function in relation
to the set-theoretic paradoxes. There can be no function taking every set


Thomas Forster, Set Theory with a Universal Set: Exploring an Untyped Universe (Oxford:
Clarendon Press, ), p. .
 Quine, New Foundations, and the Philosophy of Set Theory
to its singleton. If there were, we would then have T {α} = T{α}, and
T{α} = α, and thus T {α} = α, allowing the argument to go through,
thus yielding the Burali-Forti paradox.
Forster explains the situation slightly differently, observing that with
every proper initial segment of ordinals, we can associate two ordinals:
() the least ordinal not in X, call it “L(X),” and
() the order-type of X, call it “L(X).”
While L(X) is bigger than every member of X, this is not obviously true
of L(X). To derive the Burali-Forti paradox, we need to show by induc-
tion on the end-extension relation that (∀X)(L(X) = L(X)). But in NF,
the ordinals L(X) and L(X)) belong to different types. Hence (∀X)
(L(X) = L(X)) is unstratified, so the class of counterexamples is not
guaranteed to be a set as is required for our induction. In fact, it turns
out that there are initial segments of X of the ordinals where L(X) ≠
L(X), so no paradox arises.
As Rosser concludes in his own earlier discussion, “[I]t seems to be
the case that there is no danger of deriving a contradiction along any of
the known lines until one can handle unstratified relations more effec-
tively.” This echoes Quine’s point made earlier: With regard to consis-
tency, we are in no better position if we use other versions of set theory.
We do not, in general, have absolute consistency results for set theory,
so our confidence in its consistency lies in our being unable to derive the
paradoxes according to the usual known methods. Consistency for set
theory – any set theory – rests upon our not yet having found a means
for deriving the set-theoretic paradoxes.

. Some Concluding Remarks on the Significance of

Consistency Proofs
The line of thought on the place of consistency proofs for set theory
deserves further development. Quine and Rosser’s point about NF is not
that consistency proofs of any sort are uninteresting or unimportant for
foundational studies, and this holds both in the specific case of NF and in

Forster, Set Theory with a Universal Set, pp. –.

Rosser, “Consistency”, p. .

This is not limited to set theory. By Gödel’s second incompleteness theorem, no system strong
enough for arithmetic or, more specifically, Robinson’s Q – Peano arithmetic without the axiom
schema of induction – can prove its own consistency. In this sense, consistency proofs for such
systems will always be relative to the system in which the consistency proof is carried out, and this
system must be at least as strong as the system of which it proves the consistency.
 New Foundations & Beginnings of Quine’s Philosophy 
the more general case of mathematics as a whole. Their point is only that
since other set theories also lack consistency proofs, NF is no worse off in
this respect. To make their claim more precise, let me first consider the
sense in which consistency proofs are interesting and important to founda-
tional studies. I will then turn to consider more carefully in this context the
sense in which relative consistency proofs for set theory (and this point can
be extended to any other branch of mathematics) can serve important
mathematical and philosophical aims. Let me first consider some important
studies by Georg Kreisel and Michael Resnik on these issues.
In his  “On the Mathematical Significance of Consistency
Proofs,” Kreisel argues for the importance of consistency proofs for the
foundations of mathematics. He sets his discussion in the context of
Hilbert’s program as outlined in Volume I of Hilbert and Bernays’s
Grundlagen der Mathematik of providing a consistency proof for arith-
metic. Kreisel correctly describes this project not as one of traditional
foundationalist epistemology, aiming to provide certainty to mathematics
through something like self-evident axioms, but as epistemological never-
theless in that the means used for the proof should be “finitistic,” or
“more ‘evident’ or more constructive than the principles under discus-
sion.” However, he finds two central problems with the project: that
they leave unclear exactly what is meant by “constructive” and that the
most interesting work found in Hilbert and Bernays’s Volume II goes
well beyond just a concern with consistency. These defects bring
Kreisel to a different program, which he thinks will show more exactly
where the mathematical significance of work on consistency problems
lies. He resolves the first problem rather quickly by identifying the
notion of “constructive” with “recursive.” We could of course reject
characterizing “constructive” in this way, but I take this as no objection
to Kreisel’s program. He aims here only to make precise what he found
imprecise about Hilbert’s original notion; others are free to do likewise.
Having made clear what he will mean by constructive methods,
Kreisel then states that the real mathematical gain will be in keeping
track of the constructive content contained within mathematical concepts
and proofs that are overall non-constructive, particularly within


Georg Kreisel, “On the Mathematical Significance of Consistency Proofs,” The Journal of Symbolic
Logic : (), pp. –.

David Hilbert and Paul Bernays, Grundlagen der Mathematik,  vols., nd edn (New York, NY:
Springer-Verlag, /).

Kreisel, “On the Mathematical Significance of Consistency Proofs,” p. .

Ibid.
 Quine, New Foundations, and the Philosophy of Set Theory
arithmetic and analysis. The purpose here is not in that constructive con-
sistency proofs will yield greater evidence or reliability to mathematics in
contrast to their nonconstructive counterparts, “but in this: they help us
to keep track of the constructive (recursive) content of the steps in the
(non-constructive) proofs of the system considered. As a result one not
only understands ‘ordinary’ mathematics better but obtains new theo-
rems.” Kreisel goes on to provide examples of the sort of new theorems
that might be obtained, but the real point of emphasis in the context
of Quine’s philosophy of set theory is on furthering understanding,
as Quine takes this to be a central aim of science generally and so also of
philosophy. We will get some indication of this at the conclusion of
this chapter and will continue to emphasize this aspect of Quine’s philo-
sophy throughout Chapters  and . But let me first comment on
Resnik’s more philosophical account of consistency proofs.
Much like Kreisel’s article, Resnik’s  “On the Philosophical
Significance of Consistency Proofs” makes its aim clear in its title.
Resnik begins from the standpoint of more traditional foundationalist
approaches to the epistemology of mathematics, stating that much of the
foundational work in mathematics was initiated in response to the
set-theoretic paradoxes. Consistency proofs of the various foundational
systems were thought to contribute to this work by justifying these sys-
tems as well as the core of classical analysis and set theory. The mathe-
matical contributions of such research, he says, are uncontroversial with
Herbrand’s work, leading to the completeness proof for quantificational
logic being a prime example. However, the philosophical contributions
as a response to skeptical worries about mathematics are far less clear.
Here, the focus has typically been on absolute consistency proofs, that is,
proofs of consistency that do not depend on the consistency of some
other system. These latter proofs are the relative consistency proofs, such
as those discussed earlier in this chapter with regard to set theory. The
basic idea for absolute consistency is to reason directly about the rules of
the system itself so as to show that we could never prove from those rules
both a statement p and its negation not-p, although such a proof will still

Ibid., p. .

W. V. Quine, Pursuit of Truth, rev. edn (Cambridge, MA: Harvard University Press, ), p. .

Michael D. Resnik, “On the Philosophical Significance of Consistency Proofs,” Journal of
Philosophical Logic :/ (), pp. –.

The relevant papers of Herbrand are his  “Investigations in Proof Theory: The Properties of
True Propositions” and his  “On the Consistency of Arithmetic.” Gödel, of course, gave the
completeness proof itself in his  “The Completeness of the Axioms of the Functional
Calculus of Logic.” All of these papers are reprinted in van Heijenoort’s From Frege to Gödel.
 New Foundations & Beginnings of Quine’s Philosophy 
depend upon the truth of whatever principles were used for the consis-
tency proof. Resnik says that this is why Hilbert thought that such a proof
should rely only on finitistic or constructive means, these being epistemolo-
gically less questionable than stronger methods. As is well known, how-
ever, Gödel’s second incompleteness result shows that such a consistency
proof is not available for arithmetic as a whole unless arithmetic itself is
inconsistent. What then could be said in response to the skeptic?
Here, Resnik outlines two possibilities. One possibility looks at the
consistency of some more elementary theory, such as Peano arithmetic.
Again, by Gödel’s result, there can be no consistency proof for Peano
arithmetic within Peano arithmetic itself. But it could be argued that a
consistency proof can be given in some other system that is not stronger
than, just different from, Peano arithmetic. Resnik puts forward Gerhard
Gentzen’s consistency proof for Peano arithmetic as an example of this
phenomenon, as have others. If we accept this account of Gentzen’s proof,
then we may have a response to the skeptics. The other possibility –
and I think the philosophically more interesting one – supposes that
a system S is a strong system of set theory, say, one that is at least
strong enough to reproduce within it all of ordinary mathematics. Then
there can be no convincing response to the skeptic by way of a consis-
tency proof. Any such proof would have to take place in a system S′ that
is at least as strong as the system S, so it would not convince anyone
who already questioned the consistency of S itself. What this shows is
that the skeptic’s demand cannot be met; the skeptic requires too much


Resnik, “On the Philosophical Significance of Consistency Proofs,” p. .

Resnik raises this question at the outset of “On the Philosophical Significance of Consistency
Proofs,” saying that it needs further investigation in light of some of Feferman’s work (p. )
and concludes that despite what Feferman says, Gödel’s result still shows that a Hilbertian
approach to consistency cannot be carried out (p. ). Feferman forced the reconsideration of
the effect of Gödel’s result on Hilbert’s program by presenting an alternative account of the condi-
tions that a consistency predicate must meet. Resnik concludes that Feferman’s alternative diverges
too much from our usual understanding of consistency to be a statement of the consistency of
arithmetic. The relevant paper here of Feferman’s is Solomon Feferman, “Arithmetization of
Meta-Mathematics in a General Setting,” Fundamenta Mathematicae  (), pp. –.

This account is not universally accepted. The key difference between Peano arithmetic and
Gentzen’s system is that the latter uses transfinite induction up to the ordinal ε. Many critics
have argued that at the very least, this does not seem to be in line with the sort of finitism that
Hilbert demanded.

Resnik also raises this issue with regard to the consistency of mathematics as a whole. I am not
entirely certain what he means here. Such a proof would again require some stronger system
within which the consistency proof could be given. But if we are already looking at all of mathe-
matics, then there does not seem to be any further system available. If he means only the usual
non-set-theoretic mathematics, then this would be basically the same situation as a set theory
strong enough to capture ordinary mathematics.
 Quine, New Foundations, and the Philosophy of Set Theory
from mathematics. As Resnik concludes, “We cannot be certain that our
axioms are free from contradiction and must treat them as hypotheses
which may be abandoned or modified in the face of further mathematical
experience. This attitude is taken by many foundational worker who also
go on to voice opinions about the likelihood [Resnik’s italics] that various
systems are consistent.” As with Kreisel’s view of the mathematical sig-
nificance of consistency proofs, Resnik’s assessment of their philosophical
significant is in line with Quine’s thinking on the matter. In his remark
here, Resnik espouses a form of naturalism for mathematics in that it is
only within mathematics itself that we can give a consistency proof. The
skeptic’s demand for something more is incoherent if we hold fast to our
scientific standards. What the skeptic asks for is simply impossible.
We have seen already examples of just this sort of attitude in Quine,
and I will develop it further in some detail in Chapters  and . But let
me use the topic of consistency proofs here to give some indication of
the direction that my account of Quine’s general philosophy of set the-
ory will take in the chapters that will follow. Quine is, of course, well
aware that there are consistency proofs for set theories. Indeed, by the
time of Quine’s comment with which I began this section, Gentzen had
already proved in  the consistency of the simple theory of types.
Some further comment is needed here, however. Russell did not take the
axiom of infinity as one of the official axioms of his type theory, in either
the simple or the ramified version. Instead, he assumed both this axiom
and the axiom of choice only as a hypothesis when needed. It is for this
version of type theory that Gentzen proves the consistency, so it is not a
system that is strong enough for carrying out foundational work in
mathematics.
Still, there are consistency proofs for theories that are strong enough
for doing foundational work. These cases bring us back to relative


Resnik, “On the Philosophical Significance of Consistency Proofs,” p. .

Gentzen’s proof can be found in his “The Consistency of the Simple Theory of Types” reprinted
in The Collected Papers of Gerhard Gentzen, ed. M. E. Szabo (Amsterdam: North-Holland, ),
pp. –. I do not know how readily available in the United States Gentzen’s work was in the
s. Quine was, on the whole, well informed about mathematical research in Europe during
this time, thanks to his postdoctoral travel grant. He used it to go to Europe to meet Rudolf
Carnap and to learn firsthand the most up-to-date research in the foundations of mathematics.
Quine was also an early proponent of natural deductions systems, which were due to Gentzen as
well. However, Quine explicitly stated that he learned of such systems not from Gentzen’s work
but from meeting Stanislaw Jaskowski in Warsaw in  (W. V. Quine, The Time of My Life,
Cambridge, MA: MIT Press, , p. ). Quine himself sketches such a consistency proof for
simple type theory in his Set Theory and Its Logic, rev. edn (Cambridge, MA: Harvard University
Press, ), pp. –.
 New Foundations & Beginnings of Quine’s Philosophy 
consistency proofs, in which we prove the consistency of one system rela-
tive to some other system that is at least as strong as the theory for which
we are proving consistency. As we just saw in the work of Resnik, with
such proofs, our confidence in the proof of consistency is only as great as
our confidence in the consistency of the system within which we carried
out the proof. This is again Quine’s point that there are no absolute con-
sistency results for set theory. So if we proved Peano arithmetic to be
consistent relative to some system of set theory, say, ZFC, we would
have proved it to be consistent on the assumption that ZFC is itself con-
sistent. Speaking of rather strong mathematical theories, such as the
various set theories we have been considering in these first three chapters,
Quine himself later characterized the situation as follows:
Consistency becomes more questionable, and consistency proofs become
more urgent. But they also become more difficult and less conclusive.
The more elaborate the apparatus to which a consistency proof appeals,
the more room there is for questioning the consistency of that apparatus
itself and so for questioning the conclusion that the apparatus was used to
establish. In the more speculative axiomatizations of set theory the most
we can usually aspire to in the way of a consistency proof is a proof that
one such system is consistent if another somewhat less mistrusted one is
consistent.
If we think that a consistency proof should answer to more traditional
sorts of foundational concerns over the epistemology of mathematics –
say, for example, with how we can ever know that mathematical knowl-
edge is certain – these relative consistency results will be of little use, as
Quine notes.
Much of the point of the account of set theory that I have offered so
far has been to warn against these sorts of foundationalist aspirations.
I have been presenting a historical and philosophical account of set
theory that shows it developing from within mathematics itself and not
as serving as a sort of first philosophy for mathematics. This view also leaves
room for seeing the actual importance of relative consistency results.
They are important if, as Kreisel urged, we seek understanding. Indeed,


Again, there are other methods for showing Peano arithmetic to be consistent. Gentzen proved
Peano arithmetic to be consistent by using transfinite induction up to the ordinal ε. These methods
must also in some sense go beyond what is available in Peano arithmetic itself. Otherwise, we would
violate Gödel’s second incompleteness theorem; see his “On Formally Undecidable Propositions of
Principia Mathematica and Related Systems I,” in Collected Works. For Gentzen’s proof, see his “The
Consistency of Elementary Number Theory,” in his Collected Papers, pp. –.

Quine, Set Theory and Its Logic, p. .
 Quine, New Foundations, and the Philosophy of Set Theory
Quine takes this to be not just a mathematical aim but a central aim for
philosophy generally. Relative consistency results contribute to it in
various ways, as Kreisel noted. Quine describes others. For one, they
help to bring out conceptual connections of a sort between different
mathematical theories in that consistency proofs can show us the consis-
tency strength of one theory over another. In addition, this allows us to
track the strength of the various mathematical theories we might be will-
ing to assume. Quine highlights these features in his  Set Theory
and Its Logic, observing that consistency proofs give us a precise proof-
theoretic measure of strength. As I have already remarked upon, by
Gödel’s second incompleteness theorem, a reasonably strong set theory
cannot prove its own consistency unless that set theory is itself inconsis-
tent. But, as I remarked earlier in the chapter, when we can prove the
consistency of one set theory in another set theory, then the latter theory
is the stronger. Quine goes on to explain that a method then readily
presents itself for generating a stronger set theory from any given set the-
ory. We can add to a set theory a new axiom by way of Gödel’s coding
of formulas into statements of arithmetic – what is known as Gödel
numbering – that states the consistency of the previous theory. We
might then even form a single system of this progression of axioms, but
then the original method might be continued, again yielding stronger
and stronger set theories. Quine concludes by observing that this is the
pattern of transfinite recursion: “The strength of a system of accumulated
axioms specified along these lines is limited only by the potentialities
of transfinite recursion – the ordinal strength, in turn – of the medium
of communication in which we are specifying that infinite fund of
axioms.” Throughout the remaining chapters, we will return to the
drawing out of conceptual connections, such as this one, and keeping
track of assumptions as a central aspect of Quine’s philosophy of set the-
ory and of his philosophy more generally.
We have also, as Quine alludes to, that a relative consistency proof
can help to further our confidence in a theory that has not been as well
studied as some other theory. This characterizes the current situation

Although Quine speaks of the aim of science as understanding, there is no firm boundary between
science and philosophy in his view, so this accurately characterizes much of his philosophical
undertakings as well. On this point, see his Pursuit of Truth, p. .

Quine notes in his discussion that there are other measures of strength, perhaps related, such as
ordinal strength; see his Set Theory and Its Logic, sec. .

Quine, Set Theory and Its Logic, pp. –.

Ibid., p. . He includes a footnote with further references to this method of generating stronger
and stronger set theories.
 New Foundations & Beginnings of Quine’s Philosophy 
with regard to his NF. In comparison to Zermelo’s set theory and its
variants, NF has received far less scrutiny by logicians and mathemati-
cians. If a consistency proof could be found for NF within a variant of
Zermelo’s set theory, researchers in the field would likely feel much
more comfortable with NF as part of mainstream set theory. Again, we
see here connections between theories emerging and furthering our
understanding of set theory generally. All of this, as we will continue to
see in the remaining chapters, is fully in line with a Quinean approach
to the philosophy of set theory. So now, with Quine’s NF accounted for
and some suggestion of the philosophy behind it, let us turn to consider-
ing and developing his philosophy of set theory more fully. In particular,
we will see in Chapter  that Quine’s views on the philosophy of set the-
ory are a microcosm of his philosophy as a whole.


I should stress that this was not the situation when Quine first presented NF. In the mid-s,
set-theoretic research was a much more pluralistic undertaking than it is now. Only sometime
around the s did Zermelo’s set theory emerge as dominant and become nearly synonymous
with set theory. This topic will be discussed further in Chapters  and .
 PART II

Quine, Set Theory, and Philosophy

 CHAPTER 

Quine’s Philosophy of Set Theory

In the previous chapters, I presented the development of set theory

through W. V. Quine’s New Foundations (NF). While I did not entirely
avoid philosophical discussion, I turn more directly in this chapter and
the next to philosophy – specifically to Quine’s philosophy of set theory.
In Chapters –, I emphasized to varying degrees what I have identified
as a conception of set theory as explication. In this chapter, I will
develop this approach to the philosophy of set theory in some detail, as
found in Quine’s early work. This will continue in Chapter , where I
will focus on his mature philosophy. In earlier chapters, we saw the ori-
gins of this approach to set theory in the work of set theory’s founders.
While we find definite aspects of it in the work of Georg Cantor,
Bertrand Russell, and Ernst Zermelo, Quine was really the first to fully
develop this conception as an explicit philosophical approach to set
theory. He sums up the view nicely in his  “Whitehead and the
Rise of Modern Logic.” Reflecting on the various attempts to come to
terms with the set-theoretic paradoxes, Quine observes,
But a striking circumstance is that none of these proposals, type theory
included, has an intuitive foundation. None has the backing of common
sense. Common sense is bankrupt, for it wound up in contradiction.
Deprived of his tradition, the logician has had to resort to mythmaking.


I have contrasted this approach to set theory with that of conceptual analysis, which strives to discover
a single correct conception of set theory that can somehow be drawn out of the very concept of set
itself. Most commonly, this approach is seen in attempts to derive the axioms of set theory from the
iterative conception of set. I will return to this topic in Chapters  and . In Set Theory and Its
Philosophy (New York, NY: Oxford University Press, ), Michael Potter draws a distinction
between regressive and intuitive approaches to set theory (p. ). This seems to correspond pretty clo-
sely to what I mean by the distinction between explication and conceptual analysis, respectively.

As I mentioned in the Introduction, Quine attributes the notion of explication to Rudolf
Carnap. I take it that he and Carnap would largely agree on approaching set theory as a process of
explication, but I think Quine was the one to fully develop this as a philosophical account of set
theory. In many ways, this is the project that inspired him to begin philosophy, and it remained a
significant part of his philosophical work throughout his career.


 Quine, New Foundations, and the Philosophy of Set Theory
That myth will be best that engenders a form of logic most convenient
for mathematics and the sciences; and perhaps it will become the common
sense of another generation.
Here, we get a clear and succinct statement of the key elements of
Quine’s philosophy of set theory: that set theory after the paradoxes will
be largely a conventional affair guided by pragmatic considerations of
convenience as to which set theory will best serve as a framework for
science and mathematics. Though less vividly, Quine adhered to this
view already in his  A System of Logistic, remarking that type theory
serves its purpose if it blocks the paradoxes, and it is a view that was cer-
tainly implicit in his  dissertation. In both of these works, Quine
does not see the theory of types as capturing something like the essence
of sethood; for Quine, there is no essence to be had. As is indicative of
the tradition of set theory as explication, he aimed merely to restrict set
theory enough to prevent contradiction while leaving enough strength in
place for it still to serve as a framework for the rest of mathematics and
so also for “the abstract structure of all science.” More important, this is
an attitude that became a constant in Quine’s philosophy more generally.
It is the idea that science is a theory that we construct, though as such, it
is no less real. This attitude is what he aims to convey in the oft-quoted
remark that “[t]o call a posit a posit is not to patronize it.” This idea
informs my reading of Quine throughout. I will return to it in
Chapter , where I discuss the philosophical significance of his approach
to set theory and the foundations of mathematics as aiming to clarify
and simplify the mathematical portion of our conceptual scheme. Such

W. V. Quine, “Whitehead and the Rise of Modern Logic,” in Selected Logic Papers, p. .

In the quote, Quine speaks of the best logic. During this period, he still spoke of set theory as part
of logic, but only out of piety for the usage of his predecessors Gottlob Frege, Russell, and Carnap;
see Quine’s “Reply to Hao Wang” in The Philosophy of W.V. Quine, eds. Lewis Hahn and Paul
Arthur Schilpp, expanded edn (Chicago, IL: Open Court, ), pp. –. Quine’s later insis-
tence on distinguishing set theory from first-order quantification theory does not, I think, indicate
any significant change in his philosophical views. However, pursuing this topic further here would,
take me too far from the subject matter at hand.

W. V. Quine, A System of Logistic (Cambridge, MA: Harvard University Press, ), p. . His dis-
sertation was eventually published in its original form with a preface added by Quine as The Logic of
Sequences: A Generalization of Principia Mathematica (New York, NY: Garland Publishing, ).

W. V. Quine, The Time of My Life: An Autobiography (Cambridge, MA: MIT Press, ), p. 

Quine, Word and Object, p. . I should note that the idea of myth making also appears through-
out Quine’s philosophical work and is not limited to the mathematical sciences. It is, for Quine,
an aspect of constructing scientific theories generally. See, for example, his “On What There Is,” in
From a Logical Point of View: Nine Logico-Philosophical Essays, nd rev. edn (Cambridge, MA:
Harvard University Press, ), pp. –; “Two Dogmas of Empiricism,” in From a Logical
Point of View, p. ; and “Posits and Reality,” in The Ways of Paradox and Other Essays, rev. and
enlarged edn (Cambridge, MA: Harvard University Press, ), p. .
 Quine’s Philosophy of Set Theory 
clarification and simplification are much of what Quine takes as the aim
of philosophy as a whole.
The structure of this chapter is as follows. I begin in Section . by
returning to Russell; he was, as we will see, Quine’s first important philo-
sophical influence, particularly with regard to his work in logic and the
foundations of mathematics. It is in starting from Russell, I will argue,
that Quine develops his approach to set theory as explication. Previously,
I have mostly emphasized Russell’s adherence to the view of set theory as
explication and its more pragmatic aspects. In Section ., I will discuss
other features of his philosophy that yield some tension in his view.
I will argue in Section . that Quine’s starting point in Russell is an
attempt to come to terms with these tensions by rejecting the extra-
mathematical aspects of Russell’s philosophy so as to arrive at a philoso-
phy of set theory that answers only to concerns from within mathematics
itself. In short, we have here Quine’s naturalism restricted to the philoso-
phy of set theory. I will focus in Section . on Quine’s dissertation,
“The Logic of Sequences,” and its published version, A System of Logistic.
This work develops directly out of Quine’s attempts to clarify and sim-
plify Russell’s logicism, and it is here that we see Quine working out the
tensions of Russell’s foundational work in mathematics. We also start to
get a better sense of the unique philosophical vision motivating Quine’s
work: the view that philosophy is largely concerned with the clarification
and simplification of our conceptual scheme, a topic that will receive full
treatment in Chapter . Finally, in Section . I will return to Quine’s
work in set theory up through to the period surrounding the publication
of “New Foundations for Mathematical Logic,” but in contrast to the
discussion in Chapter , the emphasis here is much more on the philoso-
phical aims that this technical work in set theory is meant to achieve.
Much of the discussion here concerns a frequently overlooked paper
from , “On the Theory of Types.” It is here that Quine’s rejection
of more metaphysical approaches to set theory emerges explicitly.


I do not emphasize it as much here, but I also think that this influence carried over to shape
Quine’s philosophical outlook as a whole. For more on this, see my “Quine, Russell, and
Naturalism: From a Logical Point of View,” Journal of the History of Philosophy : (),
pp. –. Some of what follows overlaps with the content of that paper.

W. V. Quine, “New Foundations for Mathematical Logic” with supplementary remarks. In From
a Logical Point of View, pp. –.

W. V. Quine, “On the Theory of Types,” Journal of Symbolic Logic : (), pp. –.
I have also discussed this paper in my “The Significance of Quine’s New Foundations for the
Philosophy of Set Theory,” The Monist : (), pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
. Background in Russell
In Chapter , I stressed one strand of Russell’s thought in approaching
the foundations of mathematics: that rooted in the tradition of set theory
as explication. In this section, I will emphasize a different strand: that
rooted in more traditional philosophical concerns within which Russell’s
views on the foundations of mathematic answer more to concerns out-
side of mathematics itself. With this backdrop in place, we will be able
to see more clearly the ways in which Quine moves against Russell by
urging that set theory be developed in response to concerns coming from
within mathematics and the sciences more generally.
Russell famously opens Principles of Mathematics by declaring his two-
fold aim. First, he intends to show “the proof that all pure mathematics
deal exclusively with concepts definable in terms of a very small number
of fundamental logical concepts, and that all its propositions are deduci-
ble from a very small number of fundamental logical principles.” His
second aim is “the explanation of the fundamental concepts which
mathematics accepts as indefinable,” the purpose being “to see clearly …
the entities concerned, in order that the mind may have that kind of
acquaintance with them which it has with redness or the taste of a pine-
apple.” He describes this second aim as “purely philosophical.” The
purpose of such a reduction of mathematics to logic – what has become
known as “logicism” – was not somehow to better assure us of the cer-
tainty of mathematics; rather, it was part of a complex argument against
Idealism. Russell wanted to show that, contrary to the claims of the
Idealists, at least some of our nonmetaphysical knowledge does not ulti-
mately lead to contradiction; he thought he could achieve this by way of
reducing mathematics to logic. We see here one of the most striking
and important features of Russell’s philosophy: that his technical mathe-
matical work is intimately entwined with his philosophical aims. It is a
theme to which we will return throughout this chapter to show how
Quine’s own philosophy emerges from this entanglement. We will see
Quine, in a sense, liberating logicism from many of the philosophical

Bertrand Russell, Principles of Mathematics, nd edn (London: George Allen and Unwin Ltd.,
 [first edition ]), p. xv.

Ibid.

For full details on Russell’s emergence from and reaction to Idealism, see Peter Hylton’s com-
prehensive study Russell, Idealism, and the Emergence of Analytic Philosophy (Oxford: Clarendon
Press, ) and his “Logic in Russell’s Logicism,” in his Propositions, Functions, and Analysis,
pp. –.

Hylton, Russell, Idealism, and the Emergence of Analytic Philosophy emphasizes this aspect through-
out his account of Russell’s philosophy.
 Quine’s Philosophy of Set Theory 
burdens Russell placed on it, though Quine, too, will have philosophical
aims upon which he brings technical methods to bear.
Because the British Idealists took metaphysics to be the only subject
of which there could be knowledge that is noncontradictory, all science
short of metaphysics was considered to be ultimately contradictory.
Much of their work attempted to make explicit the contradictions found
in the mathematical and natural sciences. Russell himself began in this
tradition but rejected it sometime in the late s. His  Principles
of Mathematics emerged out of work that began during his Idealist period
as part of an attempt to rework Hegel’s encyclopedia by bringing it into
line with the sciences. Russell explains in his preface to Principles that
his project had its origins in trying to cope with some difficulties in the
foundations of dynamics, which led him to consider the philosophy of
continuity and the infinite and so also the foundations of mathematics.
Because mathematics is the basis of all science, if Russell could show that
mathematics was noncontradictory, he would have shown that the
Idealists were wrong to contend that metaphysics constituted the only
noncontradictory science. The reduction to logic would show this. In
Russell’s hands, then, logicism was part of a broader project of providing
a consistent account of science generally. More specifically, it showed, in
contrast to Idealism, that a consistent account of absolute space was
ready to hand. Truth, then, did not come in degrees as the Idealists
claimed but rather could be shown to be absolute. Whether in physics,
mathematics, or metaphysics, truth was everywhere the same. Again,
Russell thought that this could be successfully carried out by showing
that mathematics ultimately reduces to logic. As we will see, though,
this depends crucially on the reduction being to something that could
plausibly be considered to be logic.
In this context, Russell’s logic was not just a technical development
that he then brought to bear on Idealism. Rather, logic itself presupposed
a particular metaphysics, a metaphysics that Russell drew directly from
G. E. Moore:
On fundamental questions of philosophy, my position, in all its chief
features, is derived from Mr G. E. Moore. I have accepted from him
the non-existential nature of propositions (except as happen to assert

For more on Russell’s Idealism and his rejection of it, see Hylton, Russell, Idealism, and the
Emergence of Analytic Philosophy; see also Nicholas Griffin, Russell’s Idealist Apprenticeship (Oxford:
Clarendon Press, ).

For further detail on this point, see Hylton, Russell, Idealism, and the Emergence of Analytic
Philosophy, pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
existence) and their independence of any knowing mind; also the pluralism
which regards the world, both that of existents and that of entities, as com-
posed of an infinite number of mutually independent entities, with relations
which are ultimate, and not reducible to adjectives of their terms or of the
whole which these compose. Before learning these views from him, I found
myself completely unable to construct any philosophy of arithmetic, whereas
their acceptance brought about an immediate liberation from a large number
of difficulties which I believe to be otherwise insuperable. The doctrines just
mentioned are, in my opinion, quite indispensable to any even tolerably satis-
factory philosophy of mathematics, as I hope the following pages will show.
This atomistic metaphysics committed to abstract objects, that is, propo-
sitions, combined with the view that truth and falsity are absolute,
Hylton labels “Platonic atomism.” According to Platonic atomism, our
knowledge of the world is direct and unmediated, a view that runs
directly contrary to that of the Idealists, who claimed that all of our
knowledge is conditioned by various conceptual structures. These
structures, they claimed, are objective, corresponding to the way the
world actually is. But this is difficult to defend. If all knowledge is
mediated, then the claim itself that the world does in fact correspond to
these conceptual structures must also be mediated by other conceptual
structures. But then the question returns: What can we say of the objec-
tivity of these other conceptual structures? And this question will con-
tinue to arise for further such conceptual structures, resulting in an
infinite regress. While the Idealists did in fact have responses to such
charges, neither Moore nor Russell saw them as sufficient to sustain the
view that such mediated knowledge could be objective knowledge. We
could not, on the basis of the Idealist view, have genuine true knowledge
of how the world really is. Against this, Russell, following Moore,
denies this central claim of Idealism and instead takes our knowledge of
the world to be direct and unmediated. Our perception of the world is
of how the world really is. Perception here should be understood in a


Russell, Principles of Mathematics, p. xviii.

My account of Russell’s metaphysics in the period of his logicism follows that of Hylton. See his
Russell, Idealism, and the Emergence of Analytic Philosophy as well as his Propositions, Functions, and
Analysis. Many of the essays in Propositions, Functions, and Analysis deal with relevant topics, but
here I follow the brief account presented in “The Theory of Descriptions,” pp. –.

As Hylton notes, in Kant’s case this account brushes over the distinction that he draws between
intuitions and concepts; see Hylton, Propositions, Functions, and Analysis, p. , fn. .

Hylton, Propositions, Functions, and Analysis, pp. –.

The views that Russell takes from Moore are put forward perhaps most clearly in Moore’s “The
Nature of Judgment,” in G. E. Moore: The Early Essays, ed. Tom Regan (Philadelphia, PA:
Temple University Press, ), pp. –.
 Quine’s Philosophy of Set Theory 
very broad sense, since both Russell and Moore thought that we could
have something like perception of ordinary everyday physical objects as
well as perception of abstract objects. Indeed, we have already seen this
in how Russell explains the very aim of his logicist reduction: “to see
clearly … the entities concerned, in order that the mind may have that
kind of acquaintance with them which it has with redness or the taste of
a pineapple” [my emphasis]. Acquaintance is the relation that gives the
mind direct unmediated access to what is outside of it.
So among the ideas that Russell takes from Moore is that of direct
unmediated access to the world, and it is in this context that propositions
enter as central to Russell’s philosophy. Returning to the previously
quoted passage, in line with the objectivity gained by this direct access to
the world, Russell then states that propositions are entities that are inde-
pendent of any knowing mind. As he understands them, propositions
are nonlinguistic, nonmental abstract entities and, roughly speaking, the
content expressed by a declarative sentence. In making a judgment, we
have acquaintance with propositions, which have among their properties
being true or false; in a true judgment, we have acquaintance with a true
proposition, and in a false one, we have acquaintance with a false propo-
sition. So again, in contrast to the Idealists, truth and falsity do not
depend on any sort of mediation by ideas in a mind but rather result
from a direct relationship with a proposition that has the requisite prop-
erty of truth or falsity.
Given that Russell’s view of propositions is atomistic, it is in this sense
that he regards the world as a plurality made up of “an infinite number
of mutually independent entities.” Propositions themselves are complex
and so can be analyzed into their basic constituents, which Russell calls
“terms.” Hylton explains that Russell views this process of analysis as
analogous to something like chemical decomposition. For example, the
proposition “Socrates is wise” can be decomposed into its two basic com-
ponents: Socrates and wisdom, since, in paradigm cases, the proposition
actually contains the objects that it is about. Russell concludes the
quoted passage by saying that he believed that only by adopting this
metaphysics could he successfully develop his philosophy of arithmetic.
It is at this point, I think, that a certain tension begins to emerge in
Russell’s philosophy with regard to his notion of analysis, and this will
be crucial to the understanding of Quine I will present. The tension
emerges between Russell’s philosophical commitments and his more

Hylton, Propositions, Functions, and Analysis, pp. –, –.
 Quine, New Foundations, and the Philosophy of Set Theory
technical mathematical approach to carrying out his logicism. On the
one hand, he takes mathematical results to be relevant to philosophical
problems and even to solve them. On the other hand, he still holds that
there is some special further task for philosophy to do beyond mathe-
matics, namely, that it acquaints us with the indefinables of mathematics.
It is here that we find the tension in his notion of analysis.
Russell describes pure mathematics in Principles of Mathematics as that
class of propositions of the form “p implies q,” where p and q are propo-
sitions each containing at least one variable, the same in each proposi-
tion, and having no constants but logical constants. The logical constants
are notions that are definable in terms of implication, the notion of such
that, the relation of a term to the class of which it is a member, the
notion of a relation, and other further notions involved in propositions
of the form just described. Russell includes the notion of truth among
these, though it is not itself a constituent of propositions. He then aims
to show that this definition suffices to account for what has traditionally
been thought of as pure mathematics. Furthermore, this definition is
not an arbitrary decision to use a common word in an uncommon signifi-
cation, but rather a precise analysis of the ideas which, more or less
unconsciously, are implied in the ordinary employment of the term. Our
method will therefore be one of analysis, and our problem may be called
philosophical – in the sense, that is to say, that we seek to pass from the
complex to the simple, from the demonstrable to its undemonstrable
premisses.
So the notion of analysis is crucial, and it is this method that will allow
us to become acquainted with the indefinables of pure mathematics “as
the necessary residue in a process of analysis.” In addition, analysis
aims to make precise common notions that had not previously been for-
mulated with the clarity and rigor demanded by science. What, then, is
this process of analysis?
On first glance, it seems that analysis is merely the technical reduction
of mathematics to logical notions. As Russell explains, it will be through
“the labours of the mathematicians themselves” that he will be able to
obtain certainty and clarity with regard to his questions about the nature
of number, infinity, space, time, and motion and about mathematical
inference itself. In reducing such questions to questions of pure logic, we


Russell, Principles of Mathematics, p. .

Ibid., p. xv.
 Quine’s Philosophy of Set Theory 
find exact knowledge about mathematics. However, this is not the whole
story. Russell continues, explaining that the philosophy of mathematics
had previously been just as controversial and unprogressive as other
branches of philosophy. Philosophy demanded a meaning for mathe-
matics, but mathematics had no answer. Now, he claims, mathematics
does have an answer “so far at least as to reduce the whole of its proposi-
tions to certain fundamental notions of logic.” Again, it seems that the
reduction of mathematics to logic answers what used to be the concerns
of philosophy. For example, it is now fairly common to think that in its
technical reduction, logicism has shown us what the natural numbers
really are. But then Russell adds, “At this point, the discussion must be
resumed by Philosophy.” Beyond the technical reduction of mathe-
matics to logic, Russell believes that there is still some further task for
philosophy to carry out. This becomes all the more apparent if we con-
sider his contrasting of two notions of definition – the philosophical and
the mathematical – which reflect respectively the two aspects of his
notion of analysis described earlier.
Russell first introduces this distinction between philosophical and
mathematical definition in discussing Giuseppe Peano’s logical work:
It is necessary to realize that definition, in mathematics, does not mean, as
in philosophy, an analysis of the ideas to be defined into constituent ideas.
This notion, in any case, is only applicable to concepts, whereas in mathe-
matics it is possible to define terms which are not concepts. Thus also
many notions are defined by symbolic logic which are not capable of
philosophical definition, since they are simple and unanalyzable.
He goes on to explain that in mathematics, we define by simply picking
out some fixed relation to a fixed term, which only one term can have.
The basic idea is that mathematical definition consists in giving necessary
and sufficient conditions for an entity to fall under a concept. Russell
then distinguishes philosophical definitions, explaining, “The point in
which this differs from philosophical definition may be elucidated by the
remark that the mathematical definition does not point out the term in
question, and that only what may be called philosophical insight reveals
which it is among all the terms there are.” So while necessary and suffi-
cient conditions are enough for mathematical definition, which brings


Ibid., pp. –.

Ibid., p. .

Ibid., pp. –, ; Hylton, Russell, Idealism, and the Emergence of Analytic Philosophy, pp. –.

Russell, Principles of Mathematics, p. .
 Quine, New Foundations, and the Philosophy of Set Theory
us back to the idea of an analysis as merely rendering an imprecise concept
precise, philosophical definition requires something more. It seems that the
mathematical definition can pick out any particular object from among
several satisfying the requisite conditions, but philosophical definition goes
deeper, telling us in some sense which objects the defined term really is or
what its true meaning is. Russell has this sense of definition in mind
when he talks of analysis as the method for carrying out his philosophical
aim of leading us to acquaintance with the objects of pure mathematics.
Such an understanding of philosophical and mathematical definitions
fits well with Russell’s later remarks in Principles of Mathematics on defin-
ing cardinal numbers. He defines the cardinal number of a class as the
class of all classes similar to the given class, where two classes are similar
if their members can be put into one-to-one correspondence. What all
similar classes have in common, then, is their cardinal number. So a
number is just a class of a particular sort. Although this definition of car-
dinal number succeeds, mathematically speaking, Russell admits some
philosophical worries about the connection between classes and predi-
cates. In a previous section, Russell had explained that a philosophical
definition is “the analysis of the idea … into constituent ideas” but that
this notion “is only applicable to concepts,” unlike definition in mathe-
matics. Here again, we have the idea of analysis as a decomposition
into simple constituents, which is, according to Russell, what a philoso-
phical definition should give us. Furthermore, it is this notion of defini-
tion that leads us to acquaintance with the real reality, a metaphysical
reality, whereas mathematical definition does not necessarily do this.
But then a philosophical difficulty remains in that Russell does not
know whether the appropriate concepts can even be found to identify
with the numbers, that is, whether, through a process of analysis, he has
truly arrived at the indefinables of mathematics. It may be that mathema-
tical definition does not necessarily lead us to the sort of acquaintance
that philosophical definition does but that it may in some cases. How,
then, do we identify such cases? Russell himself appears at least in some


Russell makes this point in discussing the analysis of betweenness in section  of Principles of
Mathematics.

Hylton makes this point in Russell, Idealism, and the Emergence of Analytic Philosophy, p. .

Russell, Principles of Mathematics, pp. –.

Russell, Principles of Mathematics, p. . Russell also repeats this notion of philosophical definition
at the outset of his discussion of cardinal numbers, remarking that “philosophically, the word defi-
nition has not, as a rule, been employed in this sense [that is, as merely satisfying necessary and
sufficient conditions]; it has, in fact, been restricted to the analysis of an idea into its constituents”
(p. ).
 Quine’s Philosophy of Set Theory 
places to believe that this concern does not really matter, at one point
remarking of the philosophical notion,
This usage is inconvenient and, I think, useless; moreover it seems to
overlook the fact that wholes are not, as a rule, determinate when their
constituents are given, but are themselves new entities …, defined, in the
mathematical sense, by certain relations to their constituents. I shall there-
fore, in future, ignore the philosophical sense, and speak only of mathe-
matical definability.
This is at first puzzling if philosophical definition is supposed to yield
the kind of acquaintance with the simple constituents of mathematics
that Russell demands of his project. His earlier remarks on philosophical
definition were in no way disparaging and seemed to carve out an impor-
tant role for philosophy. I think that what Russell intends to say here is
that philosophical definitions are useless when it comes to discussing the
technical, mathematical details of his reduction. He still thinks that there
is a role for philosophical definition, but it is in reflecting on what the
mathematical definition has accomplished that it comes into play. For
example, have we achieved acquaintance with the actual indefinables of
mathematics? But increasingly, this yields also a tension between the phi-
losophical and the mathematical. Once the mathematical definition has
presented the requisite sorts of objects for mathematics, what further ana-
lysis is there to do? What further question is there to answer? As we will
see, Russell continues to look at both notions of definition, but increas-
ingly, he takes the mathematical notion to do the work of the philoso-
phical. Yet he remains committed to the philosophical, metaphysical
foundation of his project. It is unclear what is left for the philosophical
notion of definition to do. Russell’s account of definition is not the only
place where the tension between the mathematical and the philosophical
arises. The tension between these two aspects continues into his later
adoption of the axiom of reducibility.
In Chapter , we saw evidence of Russell’s approach to set theory as
explication in his resolving of the paradoxes. But recall that as a conse-
quence of the hierarchy of types, Russell was forced to adopt the axiom
of reducibility to recover the reduction of mathematics to logic. Let me

Russell, Principles of Mathematic, pp. –.

Another such problem arising out of Russell’s distinction between the philosophical and the math-
ematical is that of the unity of the proposition. For more on this, see my “Quine, Russell, and
Naturalism.” From Quine’s perspective, there is also a fundamental lack of clarity in the inten-
sional basis for Russell’s logicism, which again lies on the philosophical rather than the mathema-
tical side of his work. I will discuss this further later in the chapter.
 Quine, New Foundations, and the Philosophy of Set Theory
develop this point a bit further here. In resolving the paradoxes, Russell
motivated the distinctions of order and the resulting types by way of his
vicious circle principle as applied to propositional functions, but he ulti-
mately justified his system by its success in doing away with the para-
doxes. This solution is both mathematical and pragmatic – mathematical
in that it yields technical resolution to the paradoxes and pragmatic in
that the criterion of success for his system is that the paradoxes are
blocked in a way that still allows him to develop his reduction of mathe-
matics to logic. I should emphasize that the solution is tied up in the
philosophical aspects of his project in that it is rooted in the metaphysics
of propositions, which he brought to bear on his Idealist opponents. In
this vein, it is notable that Russell does not give up Platonic atomism as
a philosophical position despite initially coming upon his famous para-
dox in terms of propositional functions. Furthermore, we should observe
that, in contrast to F. P. Ramsey and Quine (who, as we saw in
Chapter , both distinguished the semantic from the class paradoxes),
Russell does not opt for the simple theory of types, blocking just the set-
theoretic paradoxes, that is, blocking just those paradoxes that are rele-
vant to mathematics. He vastly complicates his universe with the system
of orders and types rather than giving up his fundamental metaphysics.
Because logic presupposed the metaphysics of propositions, a solution to
the paradoxes means eliminating the semantic and set-theoretic para-
doxes all at once.
As we saw previously, once Russell introduces these type distinctions,
he still requires some way to talk about all properties if his logic is going
to be sufficiently strong to yield mathematics. To this end, he puts for-
ward the axiom of reducibility. Recall that the axiom states that for every
propositional function, there is an extensionally equivalent predicative
propositional function, that is, an extensionally equivalent propositional
function of the lowest order compatible with its arguments. While
Russell clearly takes the existence of propositional functions as part of his
logic, reducibility is problematic in its existence claims. Logically speaking,
there seems to be no reason to assume the existence of the predicative pro-
positional functions that the axiom puts forth. It seems a fact about the
world that we could as easily reject as accept. This raises a difficulty about
the nature of logic for Russell. Indeed, one of Wittgenstein’s most persis-
tent criticisms of Russell’s logic was questioning the justification for


Alfred North Whitehead and Bertrand Russell, Principia Mathematica, vol. , nd edn
(Cambridge: Cambridge University Press, ), p. .
 Quine’s Philosophy of Set Theory 
reducibility. The absolute truth of logic functions as a crucial component
of Russell’s attack on Idealism, but it is hard to argue on purely logical
grounds that this axiom is unconditionally true. Furthermore, the axiom
does not seem to fit with any traditional characterization of logic as self-
evident, a priori, or analytic. Even taking a Quinean stance and saying that
the truths of logic are just among the truths that we hold to most firmly in
our web of belief would seem to do little for the logical status of reducibil-
ity. A crucial axiom in Russell’s logicism then fails in its logicality.
Russell recognizes this and in fact does not try to ground the axiom in
pure logic. As we have seen, in the  paper in which he first pre-
sented his theory of types, he merely appeals to the axiom’s success in
allowing for the deduction of certain key statements of arithmetic to pro-
ceed, such as mathematical induction. In Principia Mathematica, he
develops this approach further, taking it to apply generally in the case of
adopting any axiom:
The reason for accepting an axiom, as for accepting any other proposition,
is always largely inductive, namely that many propositions which are
nearly indubitable can be deduced from it, and that no equally plausible
way is known by which these propositions could be true if the axiom
were false, and nothing which is probably false can be deduced from it. If
the axiom is apparently self-evident, that only means, practically, that it is
nearly indubitable; for things have been thought to be self-evident and
have yet turned out to be false [my emphasis].
Here, Russell’s justification for reducibility is of the wholly pragmatic sort,
relying on how successful the system in which it is used captures mathe-
matics. In his preface to Principia Mathematica, he makes the same point,
reminiscent of Zermelo in his  paper (see Chapter ), stating that
“the chief reason in favor of any theory on the principles of mathematics
must always be inductive, i.e. it must lie in the fact that the theory in
question enables us to deduce ordinary mathematics” [my emphasis].
Russell develops this approach to the foundations of mathematics most


For example, see Ludwig Wittgenstein, Tractatus Logico-Philosophicus (New York, NY: Routledge,
 []), pp. .–.; see also his Cambridge Letters: Correspondence with Russell,
Keynes, Moore, Ramsey, and Sraffa, eds. Brian McGuinnes and Georg Henrik von Wright
(Malden, MA: Blackwell Publishers, ), pp. , –, –.

Russell, “Theory of Types,” pp. –.

Whitehead and Russell, Principia Mathematica, p. .

Zermelo, Ernst, “Investigations into the Foundations of Set Theory I,” in From Frege to Gödel:
A Source Book in Mathematical Logic, ed. Jean van Heijenoort, .

Ibid., p. v. Russell expresses a very similar sentiment in the earlier Principles of Mathematics,
p. xviii.
 Quine, New Foundations, and the Philosophy of Set Theory
fully in a  lecture, “The Regressive Method of Discovering the
Premises of Mathematics.” Here, he sets out to explain in what sense
comparatively difficult and obscure propositions can be the premises for
propositions, which are, on the whole, comparatively obvious, as is typi-
cally the case with the foundations of mathematics. Within this context,
he “emphasize[s] the close analogy between the methods of pure mathe-
matics and the methods of the sciences of observation.” For example, he
says that it seems absurd to prove a claim such as “ +  = ” from the
rather complex and unintuitive axioms of a logical system, such as the one
put forward by Gottlob Frege in his / Grudgesetze der
Arithmetik or the one that Russell himself was in the process of working
out at the time of this lecture. Instead, as he would later urge in Principia
Mathematica, Russell argues that we believe the premises because they
yield consequences that we already believe to be true. This, he says, is at
the heart of the inductive method and so is really no different from finding
general laws in the natural sciences. The starting point in both cases is
“a body of propositions of which we are fairly sure.” We then work back-
ward to the premises for these claims, which, though in some sense are
more complicated, are in another sense logically simpler.
Among the advantages of such an arrangement are that it allows us to
isolate the pervading element of falsehood found in any theory and that
it lends organization to our knowledge generally. For example, in com-
ing to terms with Russell’s paradox, Russell says that the regressive
method leads us to look for the least obvious among Frege’s logical pre-
mises. In doing so, we are led from making drastic revisions to Frege’s sys-
tem, for example, denying the law of contradiction, to instead looking for
some less obvious premises to revise, such as his basic law V. Still,
obviousness itself is no substitute for any kind of Cartesian indubitability.


Bertrand Russell, “The Regressive Method of Discovering the Premises of Mathematics,” in Essays
in Analysis, Lackey, pp. –. The lecture was not published until , so Quine could not
have read it as he was developing his early views. It is striking how closely Quine’s own views end
up coinciding with Russell’s as put forth in this lecture. Quine, in this sense, can be understood
as following out to their full conclusion the remarks that Russell did publish in Principia
Mathematica, which Quine read as an undergraduate.

Russell, “Regressive Method,” p. .

Gottlob Frege, Basic Laws of Arithmetic, trans. Philip A. Ebert and Marcus Rossberg (New York,
NY: Oxford University Press, ).

Ibid., pp. –. For a full account of Russell’s views on analysis, see Paul Hager, “Russell’s
Method of Analysis,” in The Cambridge Companion to Bertrand Russell, ed. Nicholas Griffin
(New York, NY: Cambridge University Press, ), pp. –.

Russell, “Regressive Method,” p. .

Ibid., p. .
 Quine’s Philosophy of Set Theory 
It always remains a matter of degree where we simply prefer to maintain
the more obvious propositions over the less obvious ones, and even those
propositions with the highest degree of obviousness might turn out false.
Furthermore, when we find that one obvious proposition is deducible
from another, both become more certain than they were in isolation. For
this reason, Russell thinks that a complicated deductive system with many
obvious parts gains a total probability of near certainty. “Thus,” he con-
cludes, “although intrinsic obviousness is the basis of every science, it is
never, in a fairly advanced science, the whole of our reason for believing
any one proposition of science.”
In Principia Mathematica, this talk of obviousness is replaced by the
notion of self-evidence, as is seen in the quote with which we began this
discussion of an inductive justification. Again, though, Russell’s talk of
self-evidence does not carry the burden of certainty that is usually attrib-
uted to it in more foundationalist approaches to epistemology. In this
sense, it differs little from walking out into the rain and declaring that it
is raining. Furthermore, Russell also observes that self-evidence does not
yield any strong justification for an axiom, since some apparently self-
evident truths have turned out to be false, specifically Frege’s basic law
V, the source of Russell’s paradox. Regardless of self-evidence, which
reducibility surely does not have, Russell qualifies the inductive reason
for accepting an axiom, that is, that it yields the desired results and no
false ones, as the chief reason for accepting it. On these grounds, reduci-
bility is perfectly acceptable as part of Russell’s logical system. But the
tension in Russell’s views on the philosophical and mathematical aims of
his project become apparent here. In the service of an argument against
Idealism, logicism needs to be a reduction to something that we can all
agree to label as logic. In particular, logic must be absolutely true.
Reducibility makes this highly problematic, since it seems just as likely
to be true as to be false. As a merely formal technical reduction of mathe-
matics to logic, however, Russell’s system succeeds, thus making available
the pragmatic justification that he takes to justify any axiom, including
reducibility. As we will see, it is this aspect of Russell that Quine latches
onto in developing a philosophy of set theory as explication. Noticeably
absent in this approach is any more traditional concerns about the nature
of logic.
I will argue later in this chapter that Quine resolves this tension
between the philosophical and the mathematical by fully adopting

Ibid., p. .
 Quine, New Foundations, and the Philosophy of Set Theory
Russell’s view that technical results can solve philosophical problems. In
doing so, though, he will let the further task that Russell claims for phi-
losophy fall by the wayside. It is in fully committing himself to the idea
that mathematics can solve philosophical questions that, I claim, we have
the origins of Quine’s naturalistic philosophy. In short, we have Quine
taking the best methods of the science of his day – in this case, those of
the new mathematical logic – and using them to address philosophical
concerns. Let me turn now directly to Quine.

. Early Developments: From “The Logic of Sequences” to

New Foundations
Quine’s  dissertation, “The Logic of Sequences,” was his first major
work in set theory. In it, he reworked roughly the first  pages of
Whitehead and Russell’s Principia Mathematica, generalizing their
account of propositional functions and relations. He begins by explaining
that he views the dissertation as allied with Principia Mathematica but
that it is more comprehensive in its generalization of Principia
Mathematica’s account of propositional functions and relations. Principia
Mathematica has a system of monadic propositional functions to do the
work of classes along with a separate theory of dyadic relations so that
parallel axioms and theorems must then be given for each of the two dis-
tinct realms. Furthermore, Russell’s system did not generalize to n-adic
relations; it is impossible to prove theorems in general for n-adic relations
without first specifying the value of n. Since dyadic relations are enough
for reducing mathematics to the system of Principia Mathematica, this is
as far as Russell goes. By instead employing a system of sequences,
Quine’s system allows for classes and relations to be treated singly rather
than as two independent realms. As a result, Quine does not need to
reprove parallel theorems for relations once he has done so for classes.
Furthermore, such sequences may be of any arbitrary length, so theorems
for relations are proved generally for any n-adicy. In addition to this,
Quine observes also that his system is more economical in primitive
ideas, though also more elaborate in its postulates. Quine also believes
that he has achieved greater elegance in his definitions. As he would later
observe, all of this went toward the philosophical project, which “aspired,


I will discuss this to some extent in the rest of this chapter and Chapter , but for an account that
is more focused on this idea, see my “Quine, Russell, and Naturalism.”
 Quine’s Philosophy of Set Theory 
like Principia, to comprehend the foundations of logic and mathematics
and hence the abstract structure of all science.”
As we have seen, the system itself is a variation on Russell’s theory of
types, though with some important differences. As with Russell in his
 paper, Quine offers no justification for adopting types beyond the
theory’s success at resolving the paradoxes while still providing enough
strength for the reduction of ordinary mathematics to logic. In line
with the approach to set theory that Quine lays out in the quote with
which we began this chapter, throughout his dissertation he presents no
justification for his reworking of Principia Mathematica beyond the ele-
gance and simplicity that he achieves. As I have already noted, one of
Quine’s central improvements over the original system of Principia
Mathematica is his generalization of it so that theorems can be proved for
relations of any n-adicy. More significant philosophical departures from
Russell are Quine’s own take on the axiom of reducibility and his exten-
sionalizing of Russell’s hierarchy of propositional functions.
In his dissertation, Quine, too, adopts the axiom of reducibility but
not as a formal postulate of his system. Instead, he says of this axiom
that “[w]e are regarding it as an informal assumption justificatory of a
certain notational convention.” Rather than taking it as a formal postu-
late, Quine recommends instead adopting it only as a prosystematic con-
vention that says that all functions that would be nonpredicative in
Principia Mathematica be reinterpreted in his system as predicative func-
tions. Still, the assumption of reducibility is needed because otherwise,
his convention of reinterpretation might yield notation that expresses
impossible notions, that is, his system would symbolize predicative func-
tions that have no entities corresponding to them. Quine claims that his
approach makes better sense of the axiom than does Russell’s. In
Principia Mathematica, the axiom does receive full symbolic expression,
but Quine explains that the status of predicative functions remains
ambiguous, since the notion is never formally defined or taken as an


Quine, Time of My Life, p. . We will see more of how this view develops in Quine’s philosophy
in discussing his Word and Object and Set Theory and Its Logic. For now, we should just note that
it was part of Quine’s philosophical project from the very start.

In the dissertation, Quine actually offers less justification than this. He does not remark on the
paradoxes at all but rather adopts the theory of types without comment and just shows the theory’s
adequacy for capturing ordinary mathematics; see The Logic of Sequences, preface, chapters XII and
XIII. In the published version of the dissertation, the  A System of Logistic, Quine does
remark on type theory as having the purpose of resolving the paradoxes in addition to showing
the theory adequate to deriving ordinary mathematics; see p. .

Quine, The Logic of Sequences, pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
explicit primitive of the system. By rendering all functions as predicative
and relegating any talk of order to prosystematic discussions of his
system, Quine eliminates any ambiguity in the status of the notion of a
predicative function and the associated notation “φ!x.”
In the published version of his dissertation, the  A System of
Logistic, reducibility receives much less discussion. Quine perhaps recog-
nized more clearly that because his system is fully extensional, all that mat-
ters is extensional equivalence among his functions. The distinction
between predicative and nonpredicative functions, then, has no role to play
in his system. Two functions with the same extension are, from the stand-
point of his system, the same function. There is no need for any talk of
differences in order. This brings us to the second significant divergence
from Russell’s system: that in both Quine’s dissertation and the published
version of it, he never strays from a completely extensional logic. From the
outset, he makes clear that his talk of propositional functions is inter-
changeable with talk of classes and relations. While Quine makes little of
this move in his dissertation, it is a significant break with Russell, who saw
his intensional propositional functions as lying at the basis of the exten-
sional classes upon which he would then construct the rest of mathematics.
That this is an important philosophical move against Russell cannot
be emphasized enough. Russell’s adoption of propositional functions as
the basis for his logic was not without thought or motivation. First,
these were the only sorts of objects that Russell took to have the requisite
generality to be properly logical. Previously, he had granted this status
also to classes, but with the discovery of the class paradox, he recognized
that he would have to subject classes to some sort of restriction. Second,
since classes could be defined contextually in terms of propositional func-
tions, unresolved questions about their nature – whether they were inten-
sional or extensional and how a class of many objects could be treated as
a single object – no longer required answers. Third, as we saw in
Chapter , Russell saw the ontological distinctions between types as flow-
ing from, and hence justified by, the distinctions between propositional
functions and individuals, which emerged from his account of

Ibid., pp. –.

Quine would also soon publish separately his views on the axiom of reducibility; see his “On the
Axiom of Reducibility,” Mind : (), pp. –.

Quine, A System of Logistic, p. .

Ibid., pp. –, –.

I follow Hylton in my brief account here. For more detail on Russell’s reasons for taking proposi-
tional functions as fundamental and the difficulties arising from doing so, see Hylton, Russell,
Idealism, and the Emergence of Analytic Philosophy, pp. –.
 Quine’s Philosophy of Set Theory 
propositions. Hence the resolution of the paradoxes and the view of the
universe as structured into a hierarchy of types rested upon taking propo-
sitional functions as primary. Russell had no similar justification for the
hierarchy if he instead took classes as basic.
Quine, of course, rejects this intensional basis for logicism and instead
treats propositional functions extensionally, identifying them with the
classes of objects they are true of: their extensions. The move comes
without immediate justification, but it is implicit in the entire aim of the
dissertation, showing that Quine’s system is adequate for yielding the
entire system of Principia Mathematica. Hence there is no need to
adopt intensional propositional functions; their extensional counterparts
are enough. These motivations become all the more apparent in the pub-
lished version of the dissertation, A System of Logistic, as Quine explains,
This assimilation of propositional functions to classes and of predication
to membership represents no actual impoverishment of logistic, but only
the elimination of useless lumber: for, as will subsequently be shown, all
theorems of PM involving function variables or predication can be proved
in the present system under the indicated manner of translation.
But why should we prefer the extensional to the intensional? Quine says
very little of the distinction in the dissertation, but again he elaborates
his motivations in A System of Logistic in discussing his extensional treat-
ment of propositions. He warns that an intensional account may force us
“to leave the terra firma of algorithmic logic and tread more metaphysical
ground.” In particular, he complains that propositions are rather diffi-
cult to pin down as identical. An extensional account, however, provides
a clear identity criterion. We can easily identify propositional functions
as the same when they have the same extensions.
This all might seem rather ad hoc in comparison to what Russell
thought a well-motivated development of the logic required for carrying
out logicism. He thought just this, reflecting later in life on Quine’s sys-
tems of logic being such that not “even the cleverest logician would have

Quine, The Logic of Sequences, pp. –.

Ibid., chapter .

Quine, A System of Logistic, p. . In Our Knowledge of the External World (New York, NY:
W. W. Norton, ), p. , Russell talks of using logical techniques to clear away “metaphysical
lumber.”

Quine, A System of Logistic, p. .

For a later and somewhat autobiographical account of Quine’s preference for extensionality, see
his “Confessions of a Confirmed Extensionalist,” in Future Pasts: The Analytic Tradition in
Twentieth-Century Philosophy, eds. Juliet Floyd and Sanford Shieh (New York, NY: Oxford
University Press, ), pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
thought of [them] if he had not known of the contradictions.” But we
have also seen Russell himself thinking in just this way about his axiom
system, justifying it inductively on grounds that it “enables us to deduce
ordinary mathematics.” It is this strand of Russell that Quine latches
onto. In adopting an extensional view of logic, Quine pushes aside diffi-
cult, if not hopeless, worries that are external to mathematics and thus
opens the way to a philosophy of mathematics that takes place entirely
within mathematics itself. For deducing ordinary mathematics, only the
extensions matter; all else is “useless lumber.” In short, we can see
Quine beginning to work out a philosophy of mathematics in which
there is no first philosophy. He elaborates on this issue further when he
comes to consider explicitly “the more philosophical side of logic.”
Having introduced propositions as sequences on purely technical
grounds, he now considers how this account plays out with regard to
a number of philosophical issues, chief among them Russell’s intensional
account of propositional functions.
Given that Russell thought only an intensional account of proposi-
tional functions could ensure the logicality of his system, Quine’s exten-
sional account is no small departure. From it, we get a very clear
indication of what Quine takes philosophy to be. He gives no argument
from the nature of set theory or logic more generally for this move but
emphasizes only the clarity that is achieved and the fending off of more
metaphysical flights. Observing that materially equivalent propositions –
those that are both true or both false – are indistinguishable within the
propositional calculus, Quine explains that Frege simply accepted that
there are only two propositions: the true and the false. The result is no
doubt strange, but Quine approves of the motivations lying behind it:
If on the other hand Frege’s conclusions be rejected, what is propositional
identity to involve over and above material equivalence? In seeking more
exacting conditions of identity than material equivalence, danger is to be
apprehended of having to leave the terra firma of algorithmic logic and
tread more metaphysical ground; the usual formal procedure, therefore, is
to avoid the question by handling material equivalence as such and never
mentioning the identity of propositions [Quine’s italics].


Russell, My Philosophical Development, p. .

Whitehead and Russell, Principia Mathematica, p. v. Russell remarks similarly in Principles of
Mathematics (see p. xviii). More specifically, as we saw earlier in this chapter and in Chapter , he
attempts to ground the axiom of reducibility in such a way.

Quine, A System of Logistic, p. .

Ibid., p. .
 Quine’s Philosophy of Set Theory 
However, Quine’s account in terms of sequences improves upon this
further by distinguishing more finely between propositions while still
remaining fully extensional:
It is a singularity of the doctrine of propositions as sequences that it give a
definite meaning to the identity of propositions which goes far beyond
the trivial condition of material equivalence, and that in so doing the doc-
trine continues to confine itself to strictly logistical territory, and, indeed,
to import nothing exceptionable to the most ardent extensionalist.
Identity of propositions is merely identity between sequences, where two
sequences x,y and z,w are identical just in case x = z and y = w. Here,
we get a much more refined account of propositional identity than that
of just material equivalence yet one that still differentiates among propo-
sitions that appear to convey the same meaning. For example, taking α
as the class of all cities now smaller than Chicago, β as the class of all
cities now larger than Paris, y as Paris, and w as Chicago, Quine explains
that the propositions α,y and β,w are materially equivalent and seem to
convey the same information but are not the same proposition. He con-
cludes, “The doctrine of propositions as sequences thus not only endows
propositional identity with a thoroughly definite and satisfactorily discri-
minative meaning, but does so quite without commerce beyond the
strictly extensional medium of mathematical logic.” Here we have what
Quine sees as the real philosophical import of his account: the clarity it
brings to logic by keeping it fully extensional. There is no recourse to
some mysterious metaphysical realm of, for example, meanings. There is
just the clear mathematical criterion for identity among sequences.
Indeed, in Quine’s own recollections, this “bid for clarity” was one of
the dissertation’s central contributions toward the philosophical project
of comprehending the abstract structure of all science. For this reason,
he saw it as a considerable improvement over Russell’s intensional basis
for logic.
Some philosophers might complain that Quine’s moves here ignore
the important philosophical issues concerning the nature of propositions.
But Quine would deny any deeper nature to propositions than what his


Ibid.

Ibid., p. .

Quine, Time of My Life, p. . For many, this might not seem a philosophical achievement worth
achieving, but for Quine, it is. The other achievements that Quine cites (strict adherence to use
and mention and economy of definition and notation) might seem similarly underwhelming for
many philosophers. For Quine, they are central to his philosophical endeavor, and this is very tell-
ing about his aims in the philosophy of set theory and in philosophy generally.
 Quine, New Foundations, and the Philosophy of Set Theory
account already provides for. Furthermore, such criticism would fail to
recognize how Quine sees technical methods as coming to bear on philo-
sophical problems. His approach might appear to be just a technical trick
applied to an otherwise deep philosophical problem. But for Quine,
such technical maneuvers are at the heart of approaching philosophical
problems, deep or otherwise. In line with what would become the natur-
alism of his mature philosophy, Quine’s account of propositions here
appeals to the best that current science has to offer in order to make
sense of the initially hazy or even mysterious notion of a proposition.
Expressing his basic agreement with Whitehead over the nonassertiveness
of propositions in contrast to judgments, Quine writes, “The doctrine of
propositions as sequences stands in striking agreement with Whitehead’s
point of view; it presents a definite technical entity fulfilling just the
demands which he makes of a proposition.” Here, we have in mind
certain conditions that propositions should satisfy. Within the context of
a logical system such as that of Principia Mathematica or of A System of
Logistic, propositions must, for example, be bearers of truth and falsity,
be manipulable in accord with the rules of the logical system, and so on.
As long as the entities that are proposed as propositions can do this, there
is no reason not to take them to be propositions. There is no deeper
essence that such entities need to capture. It is precisely this approach
that Quine would later identify as paradigmatic of philosophical analysis.
To bring us back to our general theme, we see Quine in his early
work in logic and set theory further developing what I have been calling
set theory as explication. His logical systems do not develop from what
he sees as the nature of the essence of sets but rather answer to concerns
from within the mathematical sciences. In this way, we can see his early
logical work as a precursor to his mature naturalism applied to philoso-
phy more generally.


In response to a criticism of this sort from Kripke about Quine’s criterion for ontological reduc-
tion, he writes, in “Responses,” in Theories and Things (Cambridge, MA: Harvard University
Press, ), p. :
One of Kripke’s moral precepts deplores “the tendency to propose technical criteria with the
aim of excluding approaches that one dislikes” … He notes in illustration that I adopted a cri-
terion of ontological reduction for no other reason than that it “includes well-known cases
and excludes undesired cases …” I protest that mine was expressly a quest for an objective cri-
terion agreeing with our intuitive sorting of cases. This is a proper and characteristically philo-
sophical sort of quest, so long as one knows and says what one is doing.

Quine, A System of Logistic, p. .
 Quine’s Philosophy of Set Theory 
. New Foundations as Philosophy of Set Theory
From the time of his dissertation up to the  “New Foundations for
Mathematical Logic,” Quine published roughly fifteen additional papers.
Most of these focus on the technical development of logic, although
there are two overtly philosophical papers among them: “Ontological
Remarks on the Propositional Calculus” and “Truth by Convention.”
Still, much like the dissertation and A System of Logistic, these technical
papers reflect the heart of Quine’s philosophy of set theory. We see in all
of these papers a continual striving to clarify and simplify Quine’s basic
logical system, what he would himself later describe as an attempt “to
settle on a sanest comprehensive system of logic.” For example, in his
 “Concepts of Negative Degree,” he observes, “As with imaginary
numbers, no intuitive meaning is contemplated for these concepts of
negative degree; their utility is structural simplification.” In the 
“On Derivability,” he provides an account of derivability without appeal-
ing to classes: “[I]t is the purpose of this paper to renounce all use of
class variables, relation variables, or function variables, and find an
equivalent construction in terms of weaker devices.” As part of his clar-
ificatory work, Quine always looks to what a theory must assume and
what it can get by without, as he does in this paper. While these con-
cerns might be understood as typical of any mathematical undertaking,
for Quine they are the beginnings of what he would come to see the aim
of philosophy: a striving to clarify our conceptual scheme as a whole.
Throughout this period, Quine continues to focus his attention on
the theory of types, frequently citing what he sees as the two main draw-
backs of Russell’s system: its arbitrary grammatical exclusions of other-
wise seemingly intelligible combinations of signs and the reduplication of


Quine, “Ontological Remarks on the Propositional Calculus.” Reprinted in W. V. Quine, The
Ways of Paradox and Other Essays, rev. and enlarged edn (Cambridge, MA: Harvard University
Press, ), pp. –; “Truth by Convention.” Reprinted in The Ways of Paradox and Other
Essays, pp. –.

W. V. Quine, “Autobiography of W. V. Quine,” in The Philosophy of W. V. Quine, eds. Lewis
Edwin Hahn and Paul Arthur Schilpp (La Salle, IL: Open Court), p. .

W. V. Quine, “Concepts of Negative Degree,” Proceedings of the National Academy of Sciences of
the United States of America : (), p. . Quine describes concepts of negative degree by
way of concepts with positive degree. The basic idea is to introduce the idea of n-adic relations,
where n can be less than zero. Quine take concepts to be the general term for predicates (one-
place relations) and relations of any n-adicy.

W. V. Quine, “On Derivability,” Journal of Symbolic Logic : (), p. .

For more the beginnings of Quine’s philosophy from his early logical work, see my “Quine,
Russell, and Naturalism.”
 Quine, New Foundations, and the Philosophy of Set Theory
objects at each level of the type hierarchy. Many of the papers in this
period try to come to terms with these drawbacks. Both of the  arti-
cles “A Theory of Classes Presupposing No Canons of Type” and “Set-
Theoretic Foundations for Logic” provided ways of accounting for the type
hierarchy and its associated grammatical restrictions in terms somehow
more primitive than by way of assuming a universe stratified into types.
Quine’s hope was that he could then define types contextually:
Only such locutions issue from these [contextual] definitions as would be
countenanced by the familiar theory of types. The theory of types itself,
as an index expurgatorius, therefore becomes superfluous; [fn. deleted]
the forms which it brands meaningless are already meaningless here in the
strict sense of not being defined in terms of the primitives [Quine’s
italics].
The locutions that are banned as ungrammatical are banned simply as a
result of what could be defined contextually in terms of the primitives of
the particular system that is adopted. As we saw in Chapter , Quine’s
concerns here with type theory continued into his  “New
Foundations for Mathematical Logic.” Indeed, it was again these oddities
about types that drove Quine to develop the system presented there.
While Chapter  focused mostly on the technical details of the system
and Quine’s own early development of it, let me now situate Quine’s
aims into their more philosophical setting.
As we saw, NF emerged out of Quine’s reflections on the systems of
both Russell and Zermelo, preserving something like the type restrictions
of the one theory along with the unrestricted variables of the other. In
doing this, Quine adopts as a decidedly philosophical account of set the-
ory, elements of which we saw in the work of Cantor, Russell, and
Zermelo, specifically, the idea that set theory serves to explicate the notion
of set rather than, through a process of conceptual analysis, identifying the
uniquely correct notion of a set. Central to Quine’s approach, and to this
approach in general, is to develop a set theory that takes set theory’s math-
ematical contributions as the criterion of success for the particular system
set forth. Let me here set out the general philosophical concerns that


Quine makes these points repeatedly in discussing his dissatisfaction with types; see, for example,
his “The Inception of ‘New Foundations,’” in Selected Logic Papers, pp. –.

Quine, “A Theory of Classes Presupposing No Canons of Type,” Proceedings of the National
Academy of Sciences  (), pp. –.

Ibid.

Specifically, in Chapter , we focused on Quine’s account of how Cantor’s theorem can be recon-
ciled with a universal set.
 Quine’s Philosophy of Set Theory 
Quine sees NF as developing out of and answering to. In the chapters that
follow, I will respond in more detail to more strictly mathematical con-
cerns, such as the development of arithmetic and the axiom of choice.
Quine presents very little in the way of explicit philosophical motiva-
tions in “New Foundations for Mathematical Logic” itself. Certainly,
there is the implicit concern with simplifying our theory of sets, which
then further allows us to simplify our scientific theory as a whole so as to
further our understanding of the abstract structure of science. “New
Foundations for Mathematical Logic” demonstrates this not by pursuing
some preordained notion of a set, but by developing a logic that Quine
finds “most convenient for mathematics and the sciences.” As in the
papers leading up to “New Foundations for Mathematical Logic,” Quine
highlights the oddities of Russell’s type theory: the reduplication of
objects, including the null and universal sets along with the numbers
themselves; the lack of absolute complements and reduplication of the
class algebra for each type; and, overall, the difficulties of restoring sev-
ered connections that result from this reduplication. As I explained in
Chapter , Russell’s practice was not to adhere to explicit type restric-
tions but rather to leave formulas typically ambiguous, with the under-
standing that appropriate type indexes could always be made explicit if
need be. To arrive at the system of NF, Quine follows such ambiguity
through to its natural conclusion: If the theorems and proofs of them
can be stated without making the types explicit, then there is little reason
to think of the levels that come with the theory of types as inherent to the
set-theoretic universe. Instead, Quine takes the typing restrictions as a syn-
tactic test, which he applies only to the set existence axiom schema. Here,
as we have seen, any claim of set existence must pass the test of stratifica-
tion; that is, the membership condition on the right-hand side of the
biconditional must obey the restrictions of the theory of types. This techni-
que eliminates the various oddities that Quine disliked in Russell’s original
theory. But there are still deeper philosophical motivations here that
emerge more clearly in an interesting and overlooked paper from ,
“On the Theory of Types.” Here, Quine makes clear the philosophical
implications of NF for the philosophy of set theory generally.
Again, Quine explains how a system such as his NF does away with
the oddities of reduplication found in Russell’s type theory, driving the
point home especially well by considering the status of the empty set:
This reduplication is particularly strange in the case of the null class. One
feels that classes should differ only with respect to their members, and this
is obviously not true of the various null classes. A unique null class indeed
 Quine, New Foundations, and the Philosophy of Set Theory
still seems permissible, vacuously, if we think only of the requirements
that members be alike in type. However, other requirements of type the-
ory would be violated. For example, we want the null class to be included
in each class; hence, inasmuch as it is regarded as meaningless to relate
classes of unlike types by inclusion, we need a new null class to be
included in each class of new type.
Quine also adds here that types have difficulties proving the theorem of
arithmetic that, for all finite natural numbers n, n ≠ n + . Quine
explains that we might proceed by starting with the empty set and the
universal set of any specific type, and then, by way of standard set-
theoretic operations, we can produce the class consisting of none, one,
the other, or both of these sets as elements, giving us a set of four mem-
bers. This procedure can then be repeated until, for any given n, we get
a set of at least n members, which the proof of the theorem will require.
But finding such sets will require us to ascend into higher and higher
types, so the theorem might actually be false if we do not ascend high
enough. Whitehead and Russell were, of course, aware of this difficulty
and so assumed an axiom of infinity for each type to overcome it. But
of this solution, Quine says, “Some such axiom is in any case presumably
needed for the theory of infinite numbers; but that it should be needed
for proving finite inequalities is an anomalous effect merely of the theory
of types.”
However, Quine makes clear in this article that there is a deeper philo-
sophical point to his reworking of Russell’s theory of types, a point that
is left implicit in nearly all of his other writings where he reflects on the
move from the theory of types to NF. Quine frames his discussion in
this paper by observing that the theory of types consists of two aspects,
which we should be careful to distinguish – an ontological, or metaphysi-
cal, doctrine and a formal restriction – and it is the first of these that
Quine sees as the source of the unnatural effects of type theory. He
explains this ontological, or metaphysical, aspect as
stipulat[ing] that if an individual is a member of a class x, then x must be
composed exclusively of individuals; if an individual is a member of a
member of a class x, then x must be composed exclusively of classes com-
posed exclusively of individuals; and so on. Individuals are said to be of
type , and classes of objects of type n are said to be of type n + ; and in


Quine, “On the Theory of Types,” p. .

Whitehead and Russell, Principia Mathematica, vol. , pp. –.

Quine, “On the Theory of Types,” p. .

Ibid., pp. , .
 Quine’s Philosophy of Set Theory 
these terms the theory of types amounts, in its ontological aspect, to
demanding that all the members of a class be alike with respect to type.
I take it as significant that Quine describes this aspect as metaphysical,
since his use of this term during this period tends to follow Rudolf
Carnap’s with all of its negative connotations. Given the peculiarities
that result from the adoption of types along with the use of typical ambi-
guity in actual practice, Quine sees something extramathematical and so,
more generally, extrascientific in the commitment to types as an ontolo-
gical doctrine, asserting that the set-theoretic universe actually is stratified
into the levels of the type hierarchy. His point becomes all the more clear
in comparing what he has to say about the second aspect of type theory,
that is, its formal restriction.
According to this formal aspect, a formula is meaningless rather than
false if it includes any subformula in which the membership relation is
flanked by two variables that are not of consecutive ascending types or if
the subset or identity relation is flanked by variables that are not of the
same type. The meaningful atomic formulas then take the following
forms, respectively, xn ∈ xn+, xn ⊂ yn, or xn = yn. While often
expressed by reference to the type ontology, the formal aspect does not
dictate that the set-theoretic universe be in fact stratified into such a hier-
archy. This emerges clearly when Quine considers Russell’s actual prac-
tice of leaving the formulas typically ambiguous. As we have seen
already, Russell dispenses with any type indices, as “is more usual and
more convenient,” leaving the variables “typically ambiguous.” The idea
then is that indices could be added to the variables to show that the for-
mulas actually do adhere to the type restrictions, but there need be no
talk of the set-theoretic universe actually being stratified into levels.
Quine explains by describing the actual proof that Whitehead and


Ibid., p. .

I would say that Quine’s latter uses of this term are most often also in the critical spirit of
Carnap, although the story becomes more complicated after his explicit break with Carnap over
analyticity and what Quine describes as the blurring of the boundaries between speculative meta-
physics and natural science. Quine’s use of the term “doctrine” may also be significant, in that he
uses this years later in “Carnap and Logical Truth” in discussing the linguistic doctrine of logical
truth, another metaphysical aspect to be purged from scientific philosophy. Quine, “Carnap and
Logical Truth,” Reprinted in W. V. Quine, The Ways of Paradox and Other Essays, rev. and
enlarged edn (Cambridge, MA: Harvard University Press, ), pp. –.

For ease of exposition, I have stated the formal aspect rather informally here, as is standard in the
literature and as Quine himself initially does (“On the Theory of Types,” p. ). He presents a
completely formal statement of it as (IV) on pp. – of “On the Theory of Types.”

Quine, “On the Theory of Types,” p. .
 Quine, New Foundations, and the Philosophy of Set Theory
Russell give of the arithmetical theorem that for all natural numbers n,
n ≠ n + . By leaving their variables typically ambiguous, Whitehead and
Russell can prove this theorem in full generality without restricting it to
any particular level of the type hierarchy and without having to add an
axiom of infinity. “Such a proof, though admitted by the apparent formal-
ism of Principia Mathematica and related systems,” Quine remarks,
seems to involve an abuse of typical ambiguity: a theorem is uncondition-
ally asserted which, judged merely on its internal structure, admits deter-
minations of type not covered by the proof. Hence Whitehead and
Russell did not choose the easy way; indeed, to avoid being deceived into
this fallacious sort of argument they even brought in a heuristic notation
of suffixes for keeping track of the range of types covered by a proof [fn.
omitted]. No such precautions were explicit in the initial formalism of
their system, and indeed it would be a matter of some complexity to
incorporate them explicitly. Obviously the abuse of typical ambiguity
would be much more convenient. Further, despite its apparent lack of
cogency, this practice seems never to yield any intrinsically undesirable
theorems.
Whitehead and Russell’s proof shows that such “undesirable theorems”
are blocked not by the ontological aspect of types but by its formal
aspect. Since their proof avoids all talk of specific types to which the vari-
ables are assigned, it does not force a commitment to any claim that the
objects of the theory actually exist only at specific levels of the hierarchy.
The formal aspect, however, still plays a key role by preventing the kind
of self-membership that would yield Russell’s paradox. So Quine
concludes,
The awkward situations thus far considered actually depend, not on the
formal aspect of type theory, but only on the ontological aspect. Let us
then try abandoning the ontological aspect altogether, retaining only the
formal restrictions: for if the theory of types is adequate at all as a safe-
guard against contradictions, it must be adequate in its formal aspect
alone.
He then adopts this formal restriction for the specifying conditions
found in set abstracts or, for NF, in the comprehension axiom schema.
With the abandonment of a typed ontology, classes can now contain
members of mixed type, for example, a class may contain as members
both individuals and classes, and even some instances of self-membership


Ibid., p. .

Ibid.
 Quine’s Philosophy of Set Theory 
are allowed, such as the fairly intuitive cases V ∈ V and Λ ∉ Λ, both of
which type theory’s ontological aspect prevented. We also gain absolute
complements rather than just complements within a specific type. In
addition, constants such as “V,” “Λ,” and the numerals cease to be typi-
cally ambiguous. Indeed, typical ambiguity is dispensed with altogether,
since there are no longer any types to be ambiguous about. This allows
Whitehead and Russell’s proof of the theorem that for all natural num-
bers n, n ≠ n +  to go through without raising any awkward questions
about the abuse of types. Only the formal aspect of types is retained with
no suggestion of any sort that the universe is actually typed into levels.
As Quine concludes, “The type ontology was at best only a graphic
representation or metaphysical rationalization of the formal restrictions;
and though some such rationalization may well be desired, it seems clear
in particular that the type ontology afforded less help than hindrance.”
The type restrictions remain merely as a syntactic test for allowable for-
mulas but do not support any stronger ontological, or metaphysical,
claim about the structure of the set-theoretic universe.
Of course, present-day set theory also takes the form of cumulative
types. We have seen this already in discussing Zermelo’s set theory and
its related systems. But as with type theory, Quine has observed oddities
here too – for example, that Zermelo’s theory also rules out a universal
set as well as absolute complements, though not in the same way that
type theory does. Something very much like the ontological doctrine
that Quine described earlier continues to hold sway in contemporary
philosophy of set theory, living on in what has become known as the
iterative conception of set, as embodied in the axioms of Zermelo’s the-
ory. But much like the situation with types, Quine would see the levels
of the iterative hierarchy, though cumulative, as having their origins in a
metaphysical view about sets rather than in the mathematical theory of
sets itself. We will discuss this in more detail in Chapter . For now, let
us see how Quine’s views on set theory mature and how they fit into
and reflect the aims of his philosophical project as a whole.


Ibid., pp. –.

Ibid., p. .

See, for example, Quine’s “The Inception of ‘New Foundations’,” p. .
 CHAPTER 

Clarifying Our Conceptual Scheme: Set Theory

and the Role of Explication

In this chapter, I turn away from Quine’s early logical work and the phi-
losophy that I have argued accompanies it. Instead, I consider a central
aspect of his more general philosophical project as he describes it in his
 Word and Object: that of simplifying and clarifying our conceptual
scheme. This is not to say that the two philosophical projects are sepa-
rate undertakings. Indeed, I think they are intimately connected, both in
the sense that Quine’s basic philosophical outlook emerges from his early
work in logic and engagement with Bertrand Russell’s work and in the
sense that the philosophy of set theory that we saw emerging in
Chapter  is a natural one to hold in light of the more general philoso-
phical outlook to which Quine is committed, as I will describe in this
chapter. In Section ., I begin by sketching the main details of Quine’s
unique philosophical project of simplifying and clarifying our conceptual
scheme, that is, of our current best scientific theory. Central to this
undertaking is his appeal to the methods of modern mathematical logic
so as to make precise many of the concepts of our science. This will
bring me to discuss in Section . the notion of explication – the clarify-
ing of some unclear expression for a particular purpose, whether scientific
or more ordinary – in Quine’s philosophy. This will show both how
the technical methods of mathematical logic can provide an explication,
though an explication does not require such methods, and how his
approach to set theory can itself be an explication. In Section ., I turn


Quine, Word and Object (Cambridge, MA: MIT Press, ).

For more on this point, see my “Quine, Russell, and Naturalism.”

The notion of a conceptual scheme may be thought of as having a broader sense than our current
best scientific theory does. This is a fair point, but Quine’s conception of science is also quite
broad; see, for example, W. V. Quine, From Stimulus to Science (Cambridge, MA: Harvard
University Press, ), p. . It may, of course, be useful to distinguish between these broader
and narrower notions in certain contexts. My account here does not rule this out, but in the rather
general account I present here, there will be no occasion to so distinguish between them.

See, for example, Quine, Word and Object, pp. –.


 Clarifying Our Conceptual Scheme 
to Quine’s mature philosophy of set theory as put forth in his  Set
Theory and Its Logic. Specifically, Section . situates this work within
the context of the broader philosophical aims described for Word and
Object. Finally, because the position I attribute to Quine bears a striking
resemblance to views that are often attributed to Rudolf Carnap, I con-
clude in Section . with some analysis of where Quine and Carnap
agree and where their important differences lie.

. Clarifying Our Conceptual Scheme

A central aim of Quine’s general philosophy is to clarify and simplify our
current best scientific theory or, more generally, our conceptual scheme.
Certainly, this is a main focus of his  philosophical masterpiece,
Word and Object. If the presentation there is selective and, in places, sche-
matic, a full and detailed version of such a project is carried out in Set
Theory and Its Logic, at least with regard to the mathematical portion of
our conceptual scheme. In examining this latter undertaking in
Section ., we will see an example of what Quine’s general philosophy
looks like in detail, and we will gain a full account of his mature philoso-
phy of set theory. But let me begin in this section with the relevant details
of the more general philosophical project as laid out in Word and Object.
Quine begins section  of Word and Object, “Aims and Claims of
Regimentation,” by remarking on some of the refinements of language
that have recommended themselves earlier in the book, among them the
tightening of boundaries for vague terms, clarifying ambiguous terms,
and adopting the notation of variables and parentheses to resolve ques-
tions of cross-reference and grouping. Such departures from ordinary lan-
guage, he explains, are in a way also part of ordinary linguistic behavior
in that when the difficulties that prompt them persist, the departures
themselves become permanent parts of ordinary language. Other such
departures are called upon only as needed. In either case, Quine
observes that his interests here have dictated departures aimed less at
facilitating communication and more at clarifying our conceptual
scheme by furthering the understanding of the referential work of our
language. While some of these departures have been of a slightly more
technical nature – for example, introducing variables and parentheses to
resolve ambiguities of cross-reference and grouping – they have all fit


Quine, Set Theory and Its Logic, rev. edn (Cambridge, MA: Harvard University Press, ).

Quine, Word and Object, pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
pretty well within the sorts of ordinary practices of linguistic clarification
that are found in everyday discourse.
But there are more technical departures too, in particular in applying
logical theory to ordinary language. In this, we find the second aspect of
Quine’s philosophical project: the simplification of our conceptual
scheme. As with clarification, parentheses are a prime example of such
simplification, and the benefits that are gained are of no small importance:
They enable us to iterate a few selfsame constructions as much as we
please instead of having continually to vary our idioms in order to keep
the grouping straight. They enable us thus to minimize our stock of basic
functions, or constructions, and the techniques needed in handling them.
They enable us to subject long expressions and short ones to a uniform
algorithm, and to argue by substitutions of long expressions for short
ones, and vice versa, without readjustments of context. But for parenth-
eses or some alternative convention [fn. omitted] yielding the foregoing
benefits, mathematics would not have come far.
The notation of modern logic – what Quine often refers to as “canonical
notation” – more generally, in his view, adds much in the way of further
simplification to theory. Here, one of the primary contributions is allowing
for uniform expression of the variety of expressions available in ordinary
language, such as “any” and “every,” so that logical theory may be more
easily applied:
Clearly it would be folly to burden logical theory with quirks of usage
that we can straighten. It is the part of strategy to keep theory simple
where we can, and then, when we want to apply the theory to particular
sentences of ordinary language, to transform those sentences into a “cano-
nical form” adapted to the theory. If we were to devise a logic of ordinary
language for direct use on sentences as they come, we would have to com-
plicate our rules of inference in sundry and unilluminating ways.
These simplifications by way of a canonical notation better allow for ease of
application of logical theory to ordinary language, in particular by allowing
for the application of an efficient algorithm of deduction so as to make
explicit logical relations of implication among sentences. But this is not all
that such a canonical notation makes possible, as Quine goes on to explain:
[T]he simplification and clarification of logical theory to which a canoni-
cal notation contributes is not only algorithmic; it is also conceptual. Each
reduction that we make in the variety of constituent constructions needed

Ibid., p. .

Ibid., pp. –.
 Clarifying Our Conceptual Scheme 
in building the sentences of science is a simplification in the structure of
the inclusive conceptual scheme of science. Each elimination of obscure
constructions of notations that we manage to achieve, by paraphrase into
more lucid elements, is a clarification of the conceptual scheme of science.
The same motives that impel scientists to seek ever simpler and clearer
theories adequate to the subject matter of their special sciences are motives
for simplification and clarification of the broader framework shared by all
the sciences. Here the objective is called philosophical, because of the
breadth of the framework concerned; but the motivation is the same. The
quest of a simplest, clearest overall pattern of canonical notation is not to
be distinguished from a quest of ultimate categories, a limning of the
most general traits of reality.
So for Quine, the simplification and clarification of our scientific theories
get at the heart of what philosophy has been traditionally thought to aim
at: a divulging of the basic features of reality. But as we would expect
from Quine, such an activity is not distinct from what science itself aims
at. The philosophical pursuit might still be distinguished from that of
working scientists but only in its greater generality. Here, we have a
prime example of the contribution that Quine thought his logical work
could offer to philosophy; both early and late, he aimed at “comprehend
[ing] … the abstract structure of all science.”
We have already seen some of how the clarification contributes to a bet-
ter understanding of the referential aspects of language, but there remains
the more general issue of ontology. A particular concern here for Quine is
what ontological commitments our conceptual scheme commits us to. He
views this as a purely technical endeavor in contrast to the many philoso-
phers who think that our ontology can be discerned through common
sense or intuition. Our ontology cannot just be read off of language as it
stands, for example, given its tendency to nominalize verbs as a mere stylis-
tic variation. Quine turns again to his canonical notation to address this
issue, observing that the objects a theory “admit[s] are precisely the objects
which we reckon to the universe of values over which the bound variables
of quantification are to be considered to range.” The quantifiers “(∀x)”
and “(∃x)” in their intended interpretations are read as “every object x such

Ibid., p. .

See, for example, Quine, Word and Object, pp. –. Hylton () describes Quine’s approach
as “metaphysics naturalized.” Peter Hylton, “Quine’s Naturalism Revisited,” in A Companion to
W.V.O. Quine, eds. Gilbert Harman and Ernie Lepore (Malden, MA: Wiley-Blackwell, ),
pp. –.

Quine, Time of My Life, p. .

Quine, “Things and Their Place in Theories,” in his Theories and Things, p. .

Quine, Word and Object, p. .
 Quine, New Foundations, and the Philosophy of Set Theory
that” and “there is an object x such that,” respectively. “The quantifiers,”
Quine explains, “are encapsulations of these specially selected, unequivo-
cally referential idioms of ordinary language. To paraphrase a sentence into
the canonical notation of quantification is, first and foremost, to make its
ontic commitment explicit, quantification being a device for talking in gen-
eral of objects.” This criterion for ontological commitment does not, of
course, tell us which objects the variables range over, that is, what objects
we should allow into our universe for the bound variables of quantification
to range over. Quine takes it to be fairly uncontroversial that physical
objects will be among the objects of science, but he also includes the
abstract objects of mathematics on grounds that science would make very
little progress without the aid of mathematics. However, given that all
mathematical objects can be rendered in terms of a theory of sets alone, he
then limits the universe to include only physical objects and sets.

. Explication
In this section, I want to turn more directly to the notion of explication
and its role in Quine’s understanding of the philosophical task of simpli-
fying and clarifying our conceptual scheme. This will lead more specifi-
cally, in Section ., to an account of Quine’s particular approach to the
philosophy of set theory as one of explication in his Set Theory and Its
Logic. It is treating set theory as an explication of the notion of set that
serves as the basis for Quine’s largely pragmatic, pluralistic, and experi-
mental approach to set theory.
Quine’s most sustained discussion of the notion of explication occurs
in section  of Word and Object, “The Ordered Pair as Philosophical
Paradigm.” While in previous sections he had examined various defec-
tive nouns, such as infinitesimals or the ideal objects of physics, and
found them to be eliminable as genuine referential fragments of lan-
guage, here he describes the opposite situation. We have a noun that is


Ibid.

On this point, see chapter  of Quine’s Word and Object. For a more succinct account of deciding
on an ontology, see section  of his “The Scope and Language of Science,” in his The Ways of
Paradox and Other Essays, rev. and enlarged edn (Cambridge, MA: Harvard University Press,
). There is a good deal to say on what Quine admits into his universe, but these issues would
take me well beyond my primary concern with set theory. For more on this topic, see Peter
Hylton’s “Quine on Reference and Ontology,” in The Cambridge Companion to Quine, ed. Roger
Gibson (New York, NY: Cambridge University Press), pp. –, as well as chapter  of
Hylton’s Quine (New York, NY: Routledge, ).

Quine, Word and Object (Cambridge, MA: MIT Press, ), sec. .
 Clarifying Our Conceptual Scheme 
in some sense defective but that is also very useful in its referential status.
As Quine explains, “[I]ts utility is found to turn on the admission of
denoted objects as values of variables of quantification. In such a case
our job is to devise interpretations for it in the term positions where, in
its defectiveness, it had not used to occur.” We must then somehow
make sense of such a noun, preserving its utility while removing its
defects. Quine appeals to the ordered pair as a particularly clear case of
just this sort of phenomenon.
Typically, we find this device in mathematics, where it allows us to
assimilate relations to classes by treating the relations as classes of ordered
pairs. Its defectiveness readily appears when we try to give an account
of what an ordered pair is. Quoting Charles Peirce’s obscure attempt to
account for the ordered pair in terms of a mental diagram, Quine
recommends instead that “We do better to face the fact that ‘ordered
pair’ is (pending added conventions) a defective noun, not at home in all
the questions and answers in which we are accustomed to imbed terms
at their full-fledged best.” He observes that mathematicians take the
single postulate
If hx; yi = hz; wi then x = z and y = w; (.)
to govern all uses required of the ordered pair. So we want a single object
that will do the work of two and that satisfies this condition. The solu-
tions, Quine observes, are many, Kazimierz Kuratowski’s rendering of
hx, yi as {{x}, {x, y}} now being among the most common. But Norbert
Wiener’s {{x}, {y, Ø}} serves the purpose equally well. It is straightforward
to show that either of these classes satisfies the above postulate. This,
Quine declares, is precisely what a philosophical analysis should do:
This construction is paradigmatic of what we are most typically up to
when in a philosophical spirit we offer an “analysis” or “explication” of
some hitherto inadequately formulated “idea” or expression. We do not
claim synonymy. We do not claim to make clear and explicit what the
user of the unclear expression had unconsciously in mind all along. We


Quine, Word and Object, p. .

Ibid.

Charles Peirce, Collected Papers. Vol. II (Cambridge, MA: Harvard University Press, ), para-
graph .

Quine, Word and Object, pp. –.

Ibid., pp. –. For a proof of how Kuratowski’s definition satisfies the required condition, see
for example, Herbert B. Enderton, Elements of Set Theory (San Diego, CA: Academic Press,
), pp. –. There are plenty of other equally good analyses of the ordered pair; Quine gives
further examples on p.  of Word and Object.
 Quine, New Foundations, and the Philosophy of Set Theory
do not expose hidden meanings, as the words “analysis” and “explication”
would suggest; we supply lacks. We fix on the particular functions of the
unclear expression that make it worth troubling about, and then devise a
substitute, clear and couched in terms to our liking, that fills those func-
tions. Beyond those conditions of partial agreement, dictated by our inter-
ests and purposes, any traits of the explicans come under the head of
“don’t-cares.”
The analysis of the ordered pair is unusual only in that the conditions of
partial agreement can be made so explicitly and simply. Other cases of
analysis will not be so straightforward, but on Quine’s account, this is
still ultimately what any such analysis is meant to accomplish.
There is no answer, then, to the question of which analysis of the
ordered pair is the correct one. Any object satisfying (.) has an equal
right to being the ordered pair. The decision is largely pragmatic, depend-
ing upon the particular context of our needs and aims. We may opt for
either of these analyses or even another. For example, if we are working
within pure number theory, we might hesitate to introduce an analysis in
terms of sets and instead construe hx, yi as, for example, x · y. This is,
in general, the situation with regard to analysis. For any such analysis, or
explication,
explication is elimination. We have, to begin with, an expression or form of
expression that is somehow troublesome. It behaves partly like a term but
not enough so, or it is vague in ways that bother us, or it puts kinks in a
theory or encourages one or another confusion. But it also serves certain
purposes that are not to be abandoned. Then we find a way of accomplish-
ing those same purposes through other channels, using other and less trou-
blesome forms of expression. The old perplexities are resolved.
In the end, any questions about what an ordered pair is are removed
when this troublesome notion is replaced by a less troublesome notion.
As we have seen, the various explications that have been offered for the


Quine, Word and Object, pp. –. “Explication” is, of course, Carnap’s terminology; see, for
example, his Meaning and Necessity: A Study in Semantics and Modal Logic, nd enlarged edn
(Chicago, IL: University of Chicago Press, ), pp. –. Part of what I hoped to have shown in
Chapter  was that Quine had this notion already in place before any serious engagement with
Carnap’s work. A more general conclusion, which I have not argued for here but will claim, is
that Quine’s and Carnap’s shared philosophical aims can be traced back to the common influence
of Russell.

Here is at least part of his rejection of the analytic/synthetic distinction. Quine just does not think
that we have any idea of what the conditions of partial agreement should be for the analysis of
analyticity. I will return to this issue in comparing Quine’s and Carnap’s views on set theory.

There are other options here, as well; see Quine, Word and Object, pp. –.

Quine, Word and Object, p. .
 Clarifying Our Conceptual Scheme 
ordered pair conflict with each other, but in Quine’s view, there is no
need to worry about what to do about the conflicts, for these are among
the “don’t-cares.” This holds generally for explications.
While Quine attributes this notion of an explication to Carnap, the
basic idea was already present in Quine’s earliest logical work, emerging
from his engagement with Russell’s logicism. We saw this in Chapter 
in Quine’s rendering of propositions as any “definite technical entity ful-
filling just the demands which [we make] of a proposition.” These
demands of a proposition are what he now identifies as the conditions of
partial agreement required by a successful explication. It is notable that
Quine brings us back to his earlier engagement with logicism by includ-
ing the reduction of arithmetic to set theory among his examples of an
explication. Here, he presents an explication as a response to the more
typically philosophical question “What is a number?” and we have
Gottlob Frege replacing these somewhat mysterious entities with entities
that are perhaps less questionable, namely, classes (though Quine cites
Russell’s Principles of Mathematics as the source of his remarks on Frege),
just as Wiener and Kuratowski did for ordered pairs., In this account,
for each number n, we identify it with the class of all n-membered classes
(the seeming circularity here can be paraphrased away). To the objection

Elsewhere, Quine makes this point with regard to worries over conflicting explanations. But this
is again to confuse explanations with explications, or analyses, in Quine’s sense. An explication is
an elimination, so we do better to think of alternative eliminations rather than conflicting explana-
tions. On this point, see his From Stimulus to Science, pp. –.

Quine, Word and Object, p. , fn. . For Carnap’s view of explication, see his Meaning and
Necessity, pp. –, and his Logical Foundations of Probability, nd edn (Chicago, IL: University of
Chicago Press, ), pp. –. For an account of Carnap’s views, see Erich Reck, “Carnapian
Explication: A Case Study and a Critique,” in Carnap’s Ideal of Explication and Naturalism, ed.
Pierre Wagner (New York, NY: Palgrave-Macmillan, ), pp. –. For a discussion of how
Quine’s understanding of an explication diverges from Carnap’s, see Martin Gustafsson, “Quine’s
Conception of Explication – And Why It Isn’t Carnap’s,” in A Companion to W. V. O. Quine,
eds. Gilbert Harman and Ernie Lepore (Malden, MA: Wiley-Blackwell, ), pp. –.

Quine, A System of Logistic, p. .

It is worth noting that Quine’s words are that the classes are “presumed to be less in question”
[my emphasis] (Word and Object, p. ). This seems to reflect something of Frege’s attitude
toward classes as being logical objects and therefore more fundamental in some sense to the num-
bers themselves. In light of the paradoxes, Quine may be acknowledging here that in being a pre-
sumption, this claim may not in fact hold true. Still, there are various grounds upon which we
could still see this explication as somehow useful. A favorite achievement of the logicist reduction,
as Quine sees it, is the unity that it provides to the various branches of mathematics. Thus, by
making apparent the interconnections among these branches, such an explication furthers our
understanding of arithmetic and mathematics more generally. For an example of this attitude in
Quine, see his “Epistemology Naturalized.” In W. V. Quine. Ontological Relativity and Other
Essays (New York, NY: Columbia University Press), p. .

Bertrand Russell, Principles of Mathematics, nd edn (London: George Allen and Unwin Ltd.,
).
 Quine, New Foundations, and the Philosophy of Set Theory
that classes have properties that differ from numbers, Quine responds
that this is just to misunderstand the point of explication:
[N]othing needs be said in rebuttal of those critics, from Peano onward,
who have rejected Frege’s version because there are things about classes of
classes that we have not been prone to say about numbers [fn. omitted].
Nothing, indeed, is more logical than to say that if numbers and classes of
classes have different properties then numbers are not classes of classes;
but what is overlooked is the point of explication.
Just as with the ordered pair, this is one of many ways of explicating num-
bers. John Von Neumann and Ernst Zermelo offered other analyses. No
two are equivalent, but all serve perfectly well as the numbers. Again, the
idea that explication is elimination is central here. There is no question as to
which analysis the numbers really are. The numbers have been eliminated
in favor of sets. As with ordered pairs, we can provide a condition that any
explication of number must satisfy. Such a condition is provided by the
notion of a progression given by Giuseppe Peano’s axioms for arithmetic,
and any objects that satisfy this will serve perfectly well as the numbers.
This attitude has been a constant of Quine’s philosophy. We saw it
with regard to propositions, where Quine showed how propositions
could be rendered in terms of the sequences of his logical system. There
was no worry here about whether these are really what propositions are.
Sequences of a certain sort turned out to fulfill just the role required of
propositions in his system. Quine’s point was that there was no further
demand to be made of them. We see the same attitude now fully devel-
oped and generalized to his philosophy as a whole and so applied also to
set theory. This is where his discussion turns next.
I think this is an accurate description of his view, but some qualifi-
cation is needed. Quine begins the discussion of sets in Word and
Object by noting that, like the troublesome infinitesimals and the ideal
objects of physics, sets are useful to our scientific theorizing.
Infinitesimals and ideal objects were found to be dispensable as objects
and were ultimately accounted for by other means, though while still
preserving the utility they provided to science. We have just seen that
they can provide an account of the natural numbers and, by way of
the ordered pair, an account of relations. They can be developed
further to provide for all of the other number systems of mathematics


Quine, Word and Object, p. .

Ibid., p. .
 Clarifying Our Conceptual Scheme 
and of all mathematical objects generally. Indeed, Quine goes on to
observe that sets can provide for any abstract object that science
demands. But even this might not be enough by itself for accepting
the sets into our ontology:
Such is the power of the notion of class to unify our abstract ontology. To
surrender this benefit and face the old abstract objects again in all their pri-
meval disorder would be a wrench, worth making if it were all. But we must
remember that the utility of classes is not limited to explication of the various
other sorts of abstract objects. The power of the notion on other counts … keeps
it in continuing demand in mathematics and elsewhere as a working notion in
its own right: not only in the protean guises of number, function, state, and all
else that it has served to explicate, but also straight. It confers a power that is not
known to be available through less objectionable channels [my emphasis].
So in addition to providing unity, sets also have a place in mathematical
research not just because they can be used to give an account of other
mathematical objects but also because they also have become, in and of
themselves, an integral part of mathematical research. Quine does not
give an example of what he has in mind here, but we might consider
that the real mathematical interest in Cantor’s discovery of set theory
was in their providing an account of the mathematics of the infinite.
This, we will see in Section ., is where Quine sees the importance of
set theory in its own right entering into mathematics.
The utility that sets provide to scientific theory, both as explications
and in their own right, cannot then be dismissed lightly. However,
Quine described them as being like infinitesimals and ideal objects not
just in their utility but also in their troublesome nature. One source of
this might be thought to be their abstractness. This is not the real worry,
though, since science will always demand abstract objects of some sort,
whether they are sets, numbers, or something else. The real source of
trouble is the one with which we are already very familiar: Set theory,
unconstrained, gives rise to contradictions. Sets as objects, however,
cannot be done away with. In this way, the situation is more like that of
ordered pairs and numbers. These objects were also found useful but
were not altogether dispensable. Despite their unclarity or mysterious
nature, their status as objects had to be somehow maintained. And expli-
cating them in terms of sets accomplished this; both ordered pairs and

Ibid., p. .

I have in mind here Quine’s remark that infinite numbers “are the business of set theory at its
most characteristic” (Set Theory and Its Logic, p. ).

Quine, Word and Object, p. .
 Quine, New Foundations, and the Philosophy of Set Theory
the numbers could be eliminated in favor of sets of certain sets. The sets
themselves, though, are not eliminated in favor of some other objects
that serve as an explication of the sets. The sets are in a sense fundamen-
tal to Quine’s ontology, though not in any traditionally more metaphysi-
cal sense. The sets are ontologically fundamental in that science must
adopt some ontology, and the utility and fruitfulness of sets are enough
to recommend them. As we saw, Quine also adopts physical objects.
Still, I think explication is the right way to describe Quine’s approach
to set theory. As I have already remarked, his discussion of set theory fol-
lows just after his account of explication here. This is much like his
remarks in the slightly later “Ontological Relativity,” where he states,
Expressions are known only by their laws, they laws of concatenation the-
ory, so that any constructs obeying those laws … are ipso facto eligible as
explications of expression. Numbers in turn are known only by their laws,
the laws of arithmetic, so that nay construct obeying those laws – certain
sets, for instance – are eligible in turn as explications of number. Sets in
turn are known only by their laws, the laws of set theory.
As in Word and Object, sets are not eliminated as numbers might be by
an explication, but as with numbers, sets are known only by their laws,
so whatever objects satisfy these laws are the sets. Quine’s approach to
set theory here follows the largely pragmatic one found in the notion of
explication. As we have seen, the initial notion of a set must be weakened
to the point of consistency while still meeting the conditions of partial
agreement for sets. These would be likely to include, for example, preser-
ving some weakened form of the comprehension axiom, that the entities
concerned be extensional, and that the theory be strong enough to pro-
vide a foundation for mathematics and an account of the mathematics of
the infinite. Within these constraints, Quine explains that
On the whole, however, one manages to salvage most of the utility that
the old class theory seemed (in happy ignorance of the paradoxes) to
afford, apart from simplicity of governing principles. Naturalness, for
whatever it is worth, is of course lost; a multitude of mutually alternative,
mutually, incompatible systems of class theory arises, each with only the
most bleakly pragmatic claims to attention.
Still, there remains a sense in which Quine’s approach does eliminate in
the way that explications typically do. Recall that an explication serves to

W. V. Quine, “Ontological Relativity,” in his Ontological Relativity and Other Essays (New York,
NY: Columbia University Press, ), p. .

Quine, Word and Object, p. .
 Clarifying Our Conceptual Scheme 
eliminate “some hitherto inadequately formulated ‘idea’ or expression.”
In the case of set theory, this inadequately formulated idea is the naïve
conception of set itself; the conception that says every predicate deter-
mines a set. This initial thought about sets is eliminated in favor of one
of the apparently consistent axiom systems for set theory.
Because the conditions of partial agreement required for an explication
do not uniquely determine any one set theory as uniquely correct, set
theory for Quine is a largely pluralistic undertaking and so is largely a
comparative one as well. This, of course, does not rule out the possibility
that some one set theory will eventually triumph over all others. Quine
was explicit about this at least as early as his  “Whitehead and the
Rise of Modern Logic,” as we saw in the quote from this essay included at
the beginning of Chapter . Even if this does happen, it will be the result
of largely pragmatic concerns rather than of someone’s having somehow
discovered the essence of sets., Let us see now how Quine’s view of set
theory as explication gives rise to the distinctive approach that he develops
in his mature philosophy of set theory.

. Quine’s Mature Philosophy of Set Theory:

Set Theory and Its Logic
In Section ., I presented at a fairly general and abstract level an
account of how Quine understands the simplification and clarification of
our conceptual scheme as a central task for philosophy. As stated at the
outset of this chapter, I take it that his mature philosophy of set theory
as put forth in his  Set Theory and Its Logic is a detailed working out
of exactly this sort of philosophical project. We saw in Section . that
an explication need only meet the conditions of partial agreement for a
particular concept. Beyond this, there is much freedom to vary the expli-
cation. Quine’s example of the ordered pair was a prime example of this.
For Quine, the notion of set is much the same; it may be explicated in
various and conflicting ways depending upon which axioms we choose to
adopt. As long as the conditions of partial agreement are satisfied, there

Ibid., p. .

Of course, set theory could also remain much like the ordered pair with no one explication ever
completely displacing the others. But even if one axiomatization of set theory does eventually win
out, this does not show that the others are without interest. They might still have value, say, in
certain limited domains, perhaps as classical mechanics does even if we consider relativity to be
the more correct and generally applicable theory.

Quine, “Whitehead and the Rise of Modern Logic”, in W. V. Quine, Selected Logic Papers,
enlarged edn (Cambridge, MA: Harvard University Press ), pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
is nothing more to demand. If Quine’s philosophy of set theory is not
focused on finding the uniquely correct account of the notion of set,
what, then, does it do? Here is where we can see the philosophical point
of Set Theory and Its Logic. In line with the philosophical project laid out
in Word and Object of simplifying and clarifying our conceptual scheme,
Quine’s philosophy of set theory is a largely a comparative endeavor,
aimed at identifying and keeping track of the assumptions that the var-
ious axiomatizations of set theory make. It is in this sense that we can
view his philosophy of set theory as a full working out of the simplifica-
tion and clarification of the mathematical portion of our conceptual
scheme. Quine aims to keep careful track of at exactly what point we
must assume the existence of sets within the mathematical portion of our
conceptual scheme and what specific assumptions of mathematical power
and ontology the various renderings of set theory make. This is the cen-
tral task for a Quinean philosophy of set theory.
To this end, Quine adopts his virtual theory of classes as the central
device for carrying out this work. This virtual theory allows for the usual
class abstract notation “{x: Fx}” for “the class of all Fs,” but it does not
allow these purported abstracts to be members of classes or to be quanti-
fied over. In this sense, these abstracts are only virtual classes rather than
real ones. However, we may still place an object on the left-hand side of
the membership relation, ∈ , before a class abstract as in, for example,
“y ∈ {x: Fx},” but this simply reduces to “Fy.” In this sense, no real the-
ory of classes is yet in play. The virtual theory is merely a partial counter-
feit theory of classes built from pure logic. Only when we allow class
abstracts as values of quantifiable variables do we have a real theory of
classes. With this move, abstracts may appear on either side of the mem-
bership relation. This virtual theory has a number of key benefits for
Quine’s project. By adhering as much as possible to the virtual theory,
he manages to show in a fairly precise manner where exactly mathematics
requires the power gained from a real theory of sets. For example, not


Throughout his work on set theory, Quine typically prefers to talk of classes rather than of sets,
treating the two terms as basically synonymous. He does not generally mean to draw the technical
distinction between sets and classes that is commonly found in the literature, in which sets are
those things that can have members and also be members in contrast to classes, or proper classes,
which can only have members. He does distinguish carefully between sets and classes when the
distinction matters, although he prefers to talk of classes and ultimate classes rather than of sets
and (proper) classes (Set Theory and Its Logic, pp. –).

We should be careful to distinguish Quine’s virtual theory from those set theories that allow for
genuine proper, or, as Quine prefers, ultimate, classes. While these theories also prohibit proper
classes from being members, they do allow proper classes as values of quantifiable variables. In
this sense, such theories are very much theories of real proper classes rather than of virtual ones.

Quine, Set Theory and Its Logic, pp. –, . I have brushed over some of the more subtle
points of the virtual theory; see his section  for the full account.
 Clarifying Our Conceptual Scheme 
until section  does he introduce his first general existence assumptions,
allowing for the existence of all classes of fewer than three members,
namely, the empty set and pair sets. This allows him to prove indiscrimi-
nately that the singleton set {α} and the pair set {α, β} exist for any
classes or class abstracts. This also allows him to prove that there are
no ultimate, or as is more typical in the literature, proper, classes in his
system. Only in section  does he make the slightly more extravagant
existence assumption of finite classes generally so as to account for math-
ematical induction. Even here, though, he does not need to posit an infi-
nite class of all finite classes. Rather he introduces an axiom schema
(.) that assures him of the existence of each individual finite class.
Quine has no general qualms about infinite sets. He will later intro-
duce them, but in keeping with his general approach, he does so only at
the point at which mathematics requires this assumption. To make such
an assumption to account for the arithmetic of finite numbers seems to
him too extravagant if it can be avoided. This is, of course, in contrast
with the more usual accounts of arithmetic, which define the natural
numbers in terms of the successor operation. For each natural number,
there will have to be a successor, and for this, there must be an infinite
class. Therefore most set theories assume from the outset an axiom of
infinity, as we have seen in both Russell’s theory of types and Zermelo’s
set theory. Quine’s trick is to instead define the natural numbers in
terms of predecessors so that, while there will have to be arbitrarily large
finite numbers, there need be no infinite set. Only when he reaches
the laws for the real numbers and the need for a least bound does the
demand for infinite sets become unavoidable. Quine then does not
hesitate to make this assumption. After all, it is the concern with infinite
classes and their relative sizes that Quine describes as the real core of set
theory. The point is only to keep track of assumptions, knowing that
we can make them, and to let them into our system as needed.


Under the virtual theory, we might have class abstracts to which no real class corresponds.

Quine, Set Theory and Its Logic, pp. –.

Ibid., pp. –.

Recall that we saw this attitude expressed already in his  paper, “On the Theory of Types.”

Quine, Set Theory and Its Logic, pp. –.

Ibid., p. xii, Ch. .

Ibid., p. . Quine notes of his theory of ordinals that they are von Neumann’s version whereas his
natural numbers are Zermelo’s. A single account of both might seem a better option, but Quine
disagrees, observing that his approach allows us to “preserve economy of assumption at the level
of natural-number theory; and so I let the natural numbers and the ordinals go their separate
ways” (Set Theory and Its Logic, pp. xii–xiii). Later in the work, he approvingly cites Zermelo’s
adoption of an axiom of infinity so as to provide an account of the real numbers and whatever
mathematics lies beyond them (Set Theory and Its Logic, p. ).
 Quine, New Foundations, and the Philosophy of Set Theory
Aside from minimizing and postponing ontological assumptions,
Quine’s virtual theory has a second but related aim. Because part of the
clarificatory aspect of his project is the comparison of the various and lar-
gely incompatible axiomatizations of set theory with their differing
assumptions of power and ontology, Quine needs some fairly neutral
ground upon which he can place this discussion. No one of these axio-
matizations can serve this purpose, since it would privilege one of them
over the others. He initially suggests that we might try a somewhat infor-
mal approach to this issue. He dismisses this attempt on grounds that
very little serious reasoning in set theory can be done without making
some substantial assumptions and that these are the kinds of assumptions
that could lead us to the various set-theoretic paradoxes. We might try
just talking around the paradoxes, but this raises a difficulty for the
novices in set theory who have already heard of them. They will be left
wondering which intuitive arguments will still hold up in light of the
paradoxes. So Quine intends his virtual theory as the basis for his desired
neutral starting point:
I handle the problem by hewing a formal line from the outset, but keep-
ing my axioms weak and reasonable and thus nearly neutral. I postpone as
best I can the topics that depend on stronger axioms; and, when these
topics have to be faced, I still postpone the stronger axioms by incorporat-
ing necessary assumptions rather as explicit hypotheses into the theorems
that require them. In this way I manage to introduce the reader at some
length to the substance of set theory without any grave breach of neutral-
ity, and yet also without resorting to studied informality or artificial pro-
traction of innocence.
Only after proceeding in this way for ten chapters does he turn to con-
sider and compare the more standard axiomatizations of set theory.
This culminating part of the work is the real core of Quine’s philoso-
phy of set theory as an exploratory and experimental undertaking.
Here, he compares and contrasts the four main systems of set theory that
are commonly found in the literature –Russell’s type theory, Zermelo’s
system, Quine’s own NF and ML, and von Neumann’s system – and
their respective and often incompatible assumptions. As we have seen,
the two initial axiomatizations of set theory, Russell’s and Zermelo’s,
both came in . Quine begins with Russell’s and its development in


Quine, Set Theory and Its Logic, pp. x–xi.

I will not discuss ML and von Neumann’s theories here, as they turn the discussion toward set
theories with classes added, a topic that is not specifically relevant to my immediate concerns.
 Clarifying Our Conceptual Scheme 
response to the paradoxes. We have already seen an account of Russell’s
theory in Chapter , so I will be brief with Quine’s account of it and
just touch on of some of its key features. Quine begins with Russell’s
attempt, following Henri Poincaré, to diagnose the paradoxes as resulting
from a single source, that of a sort of vicious circle. As is familiar by
now, Russell’s paradox results from specifying a class y by way of non-
self-membership, that is, by the condition (x)(x ∈ y ≡ x ∉ x), and then
letting the quantified variable x take y itself as a value. The basic thought
here behind diagnosing the paradoxes was that it is somehow illegitimate
to presuppose y in the range of the quantifier when specifying y itself.
Quine finds nothing especially vicious in this approach when we think
of the classes as being there from the start with particular ones only
standing in need of specification (and according to Quine, Poincaré sug-
gests no temporal aspect to the creation of classes). For example, the
most typical Yale man is specified on the basis of the average Yale scores
including his own. So while there is nothing obviously viciously circular
in impredicative specification of classes, Quine does see impredicativity
as pointing toward one way of restricting the law of comprehension so as
to pare down the universe of classes to consistency. While carefully not-
ing that it is only a metaphor, he does point to this proposal as being
less arbitrary than others in the constructional metaphor it suggests; the
classes are just those that could be generated from unspecified beginnings
over an infinite period of time by way of a membership condition men-
tioning only classes previously generated. We saw in Chapter  the
details of how Russell carries this out in terms of his theory of types and
orders. Leaving the constructional metaphor aside, this set theory has as
its characteristic feature that its universe allows a transfinite ordering of
its classes in that the membership condition for each class can be given
only in terms of classes occurring earlier in the ordering. Again,
though, Russell’s initial theory of orders is an intensional theory of pro-
positional functions, not an extensional theory of classes. Once he applies
his axiom of reducibility and contextually defines the classes in terms of
propositional functions, the original intensional theory becomes, in a
sense, superfluous. As we saw in his earliest work in set theory, with the


Quine, Set Theory and Its Logic, p. .

Ibid., pp. –. Quine goes on to describe the details of Russell’s theory in more detail, begin-
ning with his intensional basis for classes and then transitioning to the modern extensional theory
of types. Because we have already seen the relevant details in Chapter , I will here skip over most
of Quine’s discussion on these matters.
 Quine, New Foundations, and the Philosophy of Set Theory
purpose of furthering clarity, Quine recommends dropping the inten-
sional theory altogether.
While the hierarchy of types does serve to block the paradoxes, Quine
observes that the restriction imposed on the comprehension axiom is of a
rather drastic sort in that it forces a revision of logic in that type theory
is a many-sorted logic. As we have seen in various places, Russell fre-
quently passes over the specific types of the variables in favor of what he
calls typical ambiguity, a suppressing of the type indexes on the variables
of a given formula. For example, appealing to the practice of typical
ambiguity, we write the formula “(∃y)(x)(x ∈ y),” but this is not itself a
formula of type theory. It is rather like “(∃yn+)(xn)(xn ∈ yn+),” where
n is schematic for some specific type index. As an actual formula of type
theory, this might be one of “(∃y)(x)(x ∈ y),” “(∃y)(x)(x ∈ y),”
and so on. The schematic version just stands in place of an infinite num-
ber of actual formulas of type theory. So despite the suppressing of type
indexes, the formula remains one of a many-sorted logic rather than of
the familiar logic of truth-functions and quantifiers. Quine’s maxim of
minimum mutilation cautions against disrupting general logic just to
fend off the paradoxes, which are only a part of the special subject matter
of set theory. This brings Quine to consider Zermelo’s set theory with
its general variables.
Quine observes that instead of providing each variable with a specific
type index, we might just have one sort of variable and introduce a predi-
cate, “Tn,” to indicate the various types. So, for example, the formulas
“(xn)Fxn” and “(∃xn)Fxn” of type theory can now instead be written with
general variables, restricted in the usual way, as “(x)(Tnx ⊃ Fx)” or
“(∃x)(Tnx . Fx).” With general variables restored, Russell’s grammatical
restriction disappears. Whereas in Russell’s system, variables on the left-
and right-hand sides of the membership relation had to be of consecutive


Quine has made this point on a number of occasions and has criticized Russell for adopting the
intensional theory so as to then define the less obscure in terms of the more obscure. Quine him-
self observes that only the foundational portion, roughly the first  pages, of Principia
Mathematica appeals to the intensional theory of propositional functions. The rest is carried out
in terms of classes and relations-in-extension. As we saw earlier in this chapter, this is exactly what
we should expect from Russell: Russell thought the propositional functions to have more claim to
being logic than the theory of classes had. This is not to say that Quine’s criticisms miss their
mark, but we should also not ignore that Russell thought the intensional basis to be better moti-
vated for his purposes than the extensional one that Quine urges. Quine’s account of Russell’s
move from propositional functions to classes is on pp. – of Set Theory and Its Logic.

Quine, Set Theory and Its Logic, pp. –.

We will further consider this move from type theory to Zermelo’s theory in Chapter .
 Clarifying Our Conceptual Scheme 
ascending types to be meaningful, formulas that violate this restriction
are now simply false. Having done away with the restriction, Quine asks
whether Russell ever needed it at all, responding that
he did not. One sees how he was led to make it. The idea of getting the
paradoxes to turn out neither true nor false but meaningless was an invit-
ing one, and the failure of definition through vicious circularity was of a
piece with it. But his system itself, by the time he had it working, was
working independently of these ideas.
This, as we will see, is an important theme for Quine and one that he
frequently returned to, particularly in his development of NF.
Restricting the general variables by way of predicates for specific types
allows for the translation of all theorems of types with special variables
into the new theory. But Quine observes that this new system “is more
glaringly incomplete than the former, because of all the newly meaning-
ful formulas whose proof or disproof we have not provided for.”
Among these that are clearly valid for types are “x ∈ y ⊃ Tnx ≡ Tn+ y,”
“(∃y)(x)(x ∈ y ≡ x ∈ z . Fx),” and “(∃y)(x)(x ∈ y ≡ (∃w)(x ∈ w ∈ z)),”
all three of which Quine says we might add as axioms or an axiom
schemata. The latter two have exactly this status in Zermelo’s system
as his axiom schema of separation and the union axiom. In addition to
being led to Zermelo’s system by adding these additional axioms,
Quine observes additional simplifications that are suggested by the
change to general variables, such as equating the null classes for each
class type.
To gain further insight into set theory, Quine takes a comparative
approach and proposes looking at how Zermelo’s theory addresses
Russell’s paradox. As we saw, Russell’s theory avoids his paradox by pre-
venting self-membership. Zermelo’s theory, with its general variables, has
no such typing restrictions; instead, it restricts comprehension by way of
the separation schema “(∃y)(x)(x ∈ y ≡ x ∈ z . Fx)” mentioned in the
previous paragraph. Separation allows only for specification of a set as a
subset of a set z. The trick to getting Russell’s paradox is to find a z big
enough that we may specify a set of all x such that x is not a member of
itself. But Quine observes that there seems to be no way of doing so:


Quine, Set Theory and Its Logic, p. .

Ibid., p. .

Ibid., pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
“[W]e need a z big enough that {x: x ∉ x} ⊆ z'[sic], and there is no visi-
ble way to such a z. Zermelo’s protection against the paradoxes consists
essentially in eschewing too big classes.” We will consider Zermelo’s
system further in Chapter . For now, I should emphasize that while
Quine has brought out differences between Russell’s type theory and
what will become Zermelo’s theory, he has importantly stressed the
rather natural connections between the theories. Even in discussing their
distinct approaches to resolving Russell’s paradox, his overall aim is better
understanding of set theory generally. Whereas much contemporary phi-
losophy of set theory aims to privilege one axiomatization over another,
Quine’s approach is one of tolerance in the interest of furthering our
understanding of set theory more generally.
We have already given substantial consideration in Chapters  and 
to Quine’s own set theory, NF, and we will add to this discussion in
Chapters  and . Let me add to this matter here as well. In light of our
most recent discussions of Russell’s theory of types and Zermelo’s of
cumulative types, let me comment on the relevance of this aspect of set
theory to NF. We have seen how naturally type theory can be accounted
for with general variables and then how easily types can then be trans-
formed into Zermelo’s theory. In a way, what this transformation shows
is that there is no inherent need for the set-theoretic universe to be typed
into levels. One way of looking at NF, already suggested by what Quine
identifies as the formal restriction of type theory, is that NF represents a
sort of minimal assumption of the supposed set-theoretic hierarchy, in
contrast to that found in Russell’s types and in Zermelo’s cumulative
types. Like Zermelo’s theory, NF’s variables are completely general, but
Zermelo’s theory still holds on, in its own way, to the idea of the set-
theoretic universe being divided into levels. In this way, unlike Quine’s
NF, Zermelo’s theory does not completely abandon the ontological doc-
trine found in type theory. Quine’s requirement of stratification is what
is left behind when this metaphysical picture of the set-theoretic hierar-
chy is dropped. As Quine remarks, “In NF there are no types. Nor is it
required that formulas be stratified to be meaningful. Stratification is
simply an ultimate, irreducible stipulation to which a formula is to con-
form if it is to qualify as a case of ‘Fx’ in the particular axiom schema [of
comprehension] [Quine’s italics].” Again, Quine does not ignore differ-
ences, but in drawing them out, he shows how much overlap there is


Ibid., pp. –. Hallett’s Cantorian Set Theory and Limitation of Size is the standard source for
more on the history and philosophy of the limitation of size approach to set theory.

Quine, Set Theory and Its Logic, p. .
 Clarifying Our Conceptual Scheme 
between the various set theories. As much as Quine wants to stress where
these alternative axiomatizations diverge, he also goes to great lengths to
bring out their structural connections.
We might find Quine’s open-mindedness about set theory surprising,
given his usual preference for ontological parsimony and his oft-stated
commitment to classical first-order logic. Let me comment on the latter
point first. I take it that Quine thinks logic a much more settled and
uncontroversial science than set theory is. Revising logic, while in princi-
ple always an option, would be a far more drastic change to our concep-
tual scheme than is moving from one set theory to another. A change in
logic violates Quine’s maxim of minimum mutilation in a way that a
change in set theory does not, especially given Quine’s view that set the-
ory is still a far from settled discipline. As to ontological parsimony,
this is a preference that Quine has often expressed, but just as with his
commitment to classical first-order logic, it is not an unexceptionable
dogma of his philosophy. Indeed, it is much less of a commitment than
the one to classical first-order logic. Quine does think that we should not
make unnecessary ontological assumptions, but we also cannot rule out
from the start which assumptions to accept and which to reject. This will
be an ongoing process of experimentation and exploration of just the
sort that Quine presents in Set Theory and Its Logic. As we have now
seen, this process focuses largely on the investigation of the various
assumptions of strength and ontology found in the different axiomatiza-
tions of set theory. This approach furthers our understanding of set
theory as a whole:
It is generally conducive to understanding to keep track of our presupposi-
tions, in point of objects and otherwise, project by project; and to wel-
come ontological economy in connection with one project even if a more
lavish ontology is needed for the next. But it is also important to have the
less economical and more powerful mathematical theories well in hand as
engines of discovery, for swift use in unforeseen places – even though in
each such case we take pains afterward to find more economical ways of
gaining the same result.


This in itself may be a controversial claim about set theory, given the now wide acceptance of
Zermelo’s set theory. The claim would have been much less controversial during the period in
which Quine did the bulk of his work in set theory. But we might also question whether matters
are as settled as they now seem to be with regard to set theory. This will be discussed further in
Chapters  and . For a recent account of Quine’s views on logic with regard to revisability, see
Matthew Carlson, “Logic and the Structure of the Web of Belief,” Journal for the History of
Analytical Philosophy : (): https://jhaponline.org/jhap/article/view/.

Quine, Word and Object, p. .
 Quine, New Foundations, and the Philosophy of Set Theory
Understanding is, after all, for Quine, one of the central aims of science
and so also of philosophy.

. An Addendum on Quine and Carnap on

Tolerance and Set Theory
As we have seen, Quine states explicitly his agreement with Carnap over the
notion of explication. Quine urges tolerance in his set theory, taking an
exploratory and experimental approach to the subject, which might also sug-
gest agreement with Carnap. However, in his debates with Carnap over
analyticity, Quine has typically been understood as rejecting Carnap’s famed
principle of tolerance. In this final section, I want to explore more care-
fully the sense in which Quine adheres to a version of tolerance and how it
differs from Carnap’s. While bringing out their differences, however,
I would also like to emphasize that Quine and Carnap agree that the funda-
mental approach to philosophy should be a scientific one. It is precisely their
different understanding of how this should be done in such a way that the
fruitless metaphysical debates of the past are overcome that leads Carnap to
embrace the principle and Quine to reject it. Let me begin with Carnap.
As is well known, much of Carnap’s work was devoted to reshaping
philosophy to bring it into line with the sciences so that it might avoid
the “empty wrangling” of the past. He saw this as particularly evident
in the debates over the foundations of mathematics that arose in the
s between logicism, intuitionism, and formalism. Rather than taking
a position as to whether any one of these views is correct, Carnap saw
value in all of them and thought that dogmatic adherence to any one of
them was contrary to the nature of science. It is in this spirit that he
urged his principle of tolerance in his  Logical Syntax of Language:
To eliminate this standpoint [that there is a single correct language
logic], together with the pseudo-problems and wearisome controversies

Quine, Pursuit of Truth, p. .

Whether this is an accurate characterization might be questioned, as it is by Gustafsson in his
“Quine’s Conception of Explication – And Why It Isn’t Carnap’s.”

We will see that Carnap himself thought this, describing their differing views on ontology as really
just a terminological difference.

Both Thomas Ricketts and Peter Hylton see tolerance as being at the heart of Quine and Carnap’s dis-
pute over the analytic/synthetic distinction. See, for example, Thomas Ricketts, “Languages and
Calculi,” in Logical Empiricism in North America, eds. Gary L. Hardcastle and Alan W. Richardson
(Minneapolis, MN: University of Minnesota Press, ), pp. –, and Hylton, Quine , pp. –.

Rudolf Carnap, Philosophy and Logical Syntax, London: Thoemmes Press,  [originally pub-
lished ], p. .

For some indication of this view see Rudolf Carnap, “Intellectual Autobiography,” in P. A.
Schilpp, ed., The Philosophy of Rudolf Carnap (Lasalle: Open Court, ), pp. –. For his dis-
cussion of the main philosophies of mathematics, see pp. –; for his general attitude of scienti-
fic approach to philosophy, see pp. –.
 Clarifying Our Conceptual Scheme 
which arise as a result of it, is one the chief tasks of this book. In it, the
view will be maintained that we have in every respect complete liberty
with regard to the forms of language; that both the forms of construction
for sentences and the rules of transformation … may be chosen quite
arbitrarily.
The idea that is encapsulated in Carnap’s principle is that there is no
single correct language of science. As he dramatically explains later in
the book, “In logic, there are no morals. Everyone is at liberty to build
up his own logic, i.e. his own form of language, as he wishes. All that is
required of him is that, if he wishes to discuss it, he must state his
methods clearly, and give syntactical rules instead of philosophical argu-
ments” [Carnap’s italics]. Previously, languages had been constructed
by first assigning meanings to the primitive logical and mathematical
symbols. This procedure determined the logical correctness of the sen-
tences and inferences of the system of language. However, Carnap
explains that since such meanings are given in language and so are inex-
act, any consequences following from them will be similarly inexact. In
accord with his principle of tolerance, he instead urges that we choose a
set of postulates and rules arbitrarily. This choice then will determine
the meaning assigned to the primitives of the system. This method
will have particular bearing on the disputes over the foundations of
mathematics:
By this method, also, the conflict between the divergent points of view on
the problem of the foundations of mathematics disappears. For language,
in its mathematical form, can be constructed according to the preferences
of any one of the points of view represented; so that no question of justifi-
cation arises at all but only the question of the syntactical consequences to
which one or other of the choices leads, including the question of non-
contradiction.
So we have complete freedom to develop any language system we like.
Furthermore, since the adoption of the postulates and rules then deter-
mine logical correctness of the system’s sentences and inferences, there is
no question as to whether the adoption of the system itself is right or
wrong. Such questions can arise only after a language system – or what
Carnap would later call a linguistic framework – has been adopted.
The general philosophical significance of Carnap’s standpoint
emerges perhaps most clearly in his  “Empiricism, Semantics, and


Rudolf Carnap, Logical Syntax of Language, trans. Amethe Smeaton (Paterson, NJ: Littlefield,
Adams, and Company, ), pp. xiv–xv.

Ibid., p. .

Ibid., p. xv.
 Quine, New Foundations, and the Philosophy of Set Theory
Ontology.” It is here that Carnap elaborates his principle and its conse-
quences in terms of the distinction between internal and external ques-
tions. He sets his discussion within the general context of debates
between realists and nominalists over whether abstract objects should be
allowed into our scientific theories. Empiricists have tended to favor
the nominalist position of avoiding such entities to the greatest extent
possible, but science cannot get very far without some abstract objects,
specifically the objects of mathematics. Carnap’s aim is to clarify this dis-
pute and to show how scientific philosophers can adopt abstract objects
without crossing over into the more questionable realm of metaphysics.
As he explains, “[I]t will be shown that using such a language does not
imply embracing a Platonic ontology but is perfectly compatible with
empiricism and strictly scientific thinking.” It is to demonstrate this
point that he introduces his talk of linguistic frameworks and the distinc-
tion between internal and external questions.
To construct a linguistic framework, Carnap describes a situation in
which we may want to speak of a new kind of entities in our language.
These new entities will require a new way of speaking, so we will have to
introduce a set of rules that govern how we may speak of them. For
example, suppose we wanted to introduce numbers into a language that
did not include them previously. We would have to have a set of rules
governing our talk of numbers. This would include a set of axioms, such
as the Peano axioms for arithmetic, and a set of rules for logical infer-
ences from these axioms. This would constitute a linguistic framework
for numbers. Given a linguistic framework, two sorts of questions can
arise: internal and external questions. Internal questions are formulated
within a linguistic framework and concern the entities of that framework.
In the linguistic framework for numbers, we might ask whether there are
prime numbers greater than a million and attempt to find and formulate
an answer also within this framework. Similarly, within a linguistic
framework for physical objects, we might ask where a particular object
is located and then attempt to answer this question using the means pro-
vided by that framework. In both of these cases, these questions are
language relative. We can ask about numbers in the former case because
we have adopted a linguistic framework for numbers, and we can ask

Rudolf Carnap, “Empiricism, Semantics, and Ontology,” in Meaning and Necessity, nd edn
(Chicago, IL: University of Chicago Press, ).

More specifically, the concern is with whether abstract objects such as propositions and properties
should be accepted into a semantic theory.

Carnap, “Empiricism, Semantics, and Ontology,” p. .

Carnap discusses linguistic frameworks and gives various examples on pp. – of “Empiricism,
Semantics, and Ontology.”
 Clarifying Our Conceptual Scheme 
about physical objects in the latter case because we have adopted a lin-
guistic framework for physical objects. In either case, there is no question
of absolute existence of the entities involved. Questions of existence will
always depend on what language we have adopted.
But there are still questions that we might want to ask about the rea-
lity of the system of new entities as a whole. Carnap describes this sort of
inquiry as what many philosophers have meant by an ontological ques-
tion: “Many philosophers regard a question of this kind as an ontological
question which must be raised and answered before the introduction of
the new language forms. The latter introduction, they believe, is legiti-
mate only if it can be justified by an ontological insight supplying an
affirmative answer to the question of reality” [Carnap’s italics]. Such
questions as these, because they take place before a particular linguistic
framework has been adopted, are the external questions. Carnap rejects
this more metaphysical approach to philosophy; since they apparently
take place outside of any linguistic framework, they are nonsensical. But
by appealing to his linguistic frameworks, he reconstrues these traditional
questions of ontology so as to render them free of metaphysics:
We may still speak … of “the acceptance of the new entities” since this
form of speech is customary; but one must keep in mind that this phrase
does not mean for us anything more than acceptance of the new frame-
work, i.e., of the new linguistic forms. Above all, it must not be interpreted
as referring to an assumption, belief, or assertion of “the reality of the enti-
ties.” There is no such assertion. An alleged statement of the reality of the
system of entities is a pseudo-statement without cognitive content.
However, Carnap’s reconstrual still leaves these questions about the sys-
tem as a whole outside of the system itself, so what are we to say of them
from his scientific approach to philosophy? Carnap takes such questions
to be in principle different from the internal, or theoretical, questions, as
he next explains:
To be sure, we have to face at this point an important question; but it is a
practical, not a theoretical question; it is the question of whether or not to
accept the new linguistic forms. The acceptance cannot be judged as being
either true or false because it is not an assertion. It can only be judged as
being more less expedient, fruitful, conducive to the aim for which the
language is intended. Judgments of this kind supply the motivation for
the decision of accepting or rejecting the kind of entities.


Carnap, “Empiricism, Semantics, and Ontology,” p. .

Ibid.

Ibid.
 Quine, New Foundations, and the Philosophy of Set Theory
So the external questions are pragmatic in nature rather than theoretical.
They cannot be true or false because questions of truth or falsity are rela-
tive to a particular linguistic framework. Because the framework must be
adopted before such questions can arise, it makes no sense, from
Carnap’s perspective, to judge the adoption of a particular framework
as correct or incorrect. We can urge the adoption of one framework
over another only on pragmatic grounds such as simplicity, elegance, or
fruitfulness.
As I noted above, Carnap took such an approach to avoid the unpro-
gressive disputes of past philosophy. Whereas previous philosophers
appeared to reach insoluble standoffs between their various and incom-
patible views, Carnap’s method is supposed to show the way out of
these insoluble disputes by exposing the pseudo-problems that underlie
them. There is no question of whether, for example, numbers really
exist in some absolute metaphysical sense. There is only the question of
which linguistic framework to adopt. With the choice of the framework
made explicit, there is no longer any genuine dispute to be resolved. To
take an example from the foundations of mathematics, we might adopt
the comparatively weak mathematical language of intuitionism in one
setting and the comparatively strong one of classical mathematics in
another, as was Carnap’s own approach in his Logical Syntax of
Language. Depending upon our aims, the safety of the former might be
preferable to the greater ease of use and simplicity of the latter or
vice versa. More generally, we might say the same of disputes between
nominalists and realists. Again, there is no real dispute here, only a dif-
ference of linguistic frameworks, and either might be chosen depending
upon which serves our purpose better. Overcoming these pseudo-
debates with the goal of making philosophy a truly scientific and pro-
gressive discipline is at the heart of Carnap’s urging of tolerance, as he
concludes,
The acceptance or rejection of abstract linguistic forms … will finally be
decided by their efficiency as instruments, the ratio of the results
achieved to the amount and complexity of the efforts required. To decree
dogmatic prohibitions of certain linguistic forms instead of testing them
by their success or failure in practical use, is worse than futile; it is posi-
tively harmful because it may obstruct scientific progress … Let us grant
to those who work in any special field of investigation the freedom

Carnap suggests these considerations explicitly in his Foundations of Logic and Mathematics
(Chicago, IL: University of Chicago Press, ), pp. –. Quine also suggests them in his Set
Theory and Its Logic, in the final section comparing the various systems of set theory (pp. –).
 Clarifying Our Conceptual Scheme 
to use any form of expression which seems useful to them; the work
in the field will sooner or later lead to the elimination of those forms
which have no useful function. Let us be cautious in making assertions and
critical in examining them, but tolerant in permitting linguistic forms
[Carnap’s italics].
On the face of it, Quine’s approach to set theory might seem very much
like Carnap’s general attitude of tolerance. Indeed, Carnap himself
observes – citing Quine’s remark that when pursuing ontology “the
obvious counsel is tolerance and an experimental spirit” – that their
divergence over ontology is really only a terminological one. But is this
really so? Many commentators have argued that Quine’s rejection of
Carnap’s principle of tolerance is central to their differences over the
legitimacy of the analytic/synthetic distinction. Perhaps Quine takes a
different attitude toward set theory than he does toward other scientific
theories. We will see that there may be some truth in understanding
Quine in this way, but on the whole, I think that his attitude toward tol-
erance is the same for all scientific theories. After all, we just saw that he
urges tolerance in the case of ontology generally. Let us return now to
Quine so that we may see more clearly the sense in which Quine can
urge tolerance while still rejecting Carnap’s principle of tolerance.
Throughout our discussions of Quine’s philosophy of set theory, we
have seen him urging a tolerant attitude, and this is true of both his early
and late views. Recall, for instance, his remark in the  paper
“Whitehead and the Rise of Modern Logic” that none of the various set
theories “has an intuitive foundation,” so the best among them will be
that which “engenders a form of logic most convenient for mathematics
and the sciences.” As we have seen, this attitude carries through into
Quine’s mature philosophy of set theory with its emphasis on explora-
tion and experimentation and the comparison of the various set theories.
So we also cannot explain Quine’s tolerance by saying that it was an early


Carnap, “Empiricism, Semantic, and Ontology,” p. . This attitude is also evident in his earlier
Logical Syntax of Language and Foundations of Logic and Mathematics, although in those works, his
focus is specifically on the foundations of mathematics. In his “Autobiography,” Carnap makes
the point that some version of this view was always present in his thinking. Carnap, “Intellectual
Autobiography.” In Schilpp, P. A. The Philosophy of Rudolf Carnap ((La Salle, IL: Open Court,
), pp. –.

Quine, “On What There Is,” p. .

Carnap, “Empiricism, Semantics, and Ontology,” p. , fn. .

For some examples, see fn. .

Quine, “Whitehead and the Rise of Modern Logic,” p. .
 Quine, New Foundations, and the Philosophy of Set Theory
commitment that he shifted away from in his later work. How, then,
does he understand his notion of tolerance?
As we saw, Carnap’s version of tolerance is directly connected to his
notion of a linguistic framework, which allows him to render philosophical
questions free of metaphysics by reconstruing them as questions concerning
only of which linguistic framework to adopt. The crucial distinction for
Carnap here is that the choice of a linguistic framework is in principle dif-
ferent from the theoretical claims made within that framework. The former
can be evaluated only by pragmatic criteria such as expedience, fruitfulness,
and whether it suits the aims for which the linguistic framework was
intended. The latter, however, are evaluated in accordance with the logi-
cal and empirical methods provided by that framework and so can be
judged true or false. We should be tolerant toward the framework because
there can be no evaluation of it as correct or incorrect, since justification is
always relative to a particular linguistic framework. Here is precisely the
point at which Quine rejects the kind of tolerance that Carnap urges.
Instead, in Quine’s view, in both cases we are appealing to the pragmatic
criteria that Carnap says are alone operative in the choice of a linguistic fra-
mework. Quine’s holism is crucial here, that is, his view that a statement
is to be evaluated not by direct comparison with experience but rather by
looking at its contribution to the theory as a whole. For Quine, we are


However, Quine’s sympathy with Gödel’s constructible universe, V = L, may indicate a growing
preference for ZF style set theory in his later work. See, for example, Quine’s  “Immanence
and Validity” reprinted in Selected Logic Papers, p. , or his Pursuit of Truth, p. . Such a pre-
ference is in no way inconsistent with Quine’s exploratory and experimental approach. After all,
part of ongoing science is that one theory may emerge as dominant and so be taken as correct.
Quine never indicates that he thinks we have reached this point with set theory, but there is no
reason to rule out that we might be heading in that direction.

Carnap, “Empiricism, Semantics, and Ontology,” p. .

The discussion that follows in this paragraph follows Peter Hylton’s exposition in his Quine,
pp. –, as well as Hylton’s “Analyticity and the Indeterminacy of Translation,” Synthese :
(), pp. –. In the latter, Hylton notes a possible response from Carnap on this point: He
might counter that only pragmatic criteria are operative in choosing a framework, whereas both
confirmational and pragmatic criteria are operative in assessing theoretical claims. This would still
mark a difference, in principle, between the two, since assessing the theoretical claims would still be
a rule governed activity, that is, something that fulfills Carnap’s ideal of rational adjudicability in
contrast to the choice of framework. From Quine’s perspective, however, this begs the question,
since it is the status of such rules that are in question. This more complex view seems more likely
to be Carnap’s actual view, since he explicitly admits that a version of holism is operative within a
linguistic framework as well. See, for example, his Logical Syntax of Language, pp. –.

This does not mean that we need to evaluate the sentence in terms of its contribution to the
whole of science. We may look at its contribution to some subtheory. On this point, see W. V.
Quine, “Two Dogmas in Retrospect,” reprinted in W. V. Quine, Confessions of a Confirmed
Extensionalist and Other Essays, eds. Dagfinn Føllesdal and Douglas B. Quine (Cambridge, MA:
Harvard University Press, ), p. .
 Clarifying Our Conceptual Scheme 
assessing how well an entire theory allows us to cope with our sensory
experience. Thus the most abstract claims of science and mathematics,
which Carnap typically renders to the framework for science, are evaluated
in the same way as are the claims that are more directly related to sensory
evidence. That is, we accept the individual claims of a theory according to
whether they are part of a theory that handles experience overall better than
its rivals do. The theory as a whole is evaluated in terms of such features as
elegance, simplicity, and fruitfulness – just the sort of criteria that Carnap
said are operative at the level of choosing a linguistic framework. So for
Quine, in contrast to Carnap, there is no real difference between how we
evaluate the choice of the linguistic framework and how we evaluate the
choice of a theory within that framework. Justification, then, is not a
language-relative matter.
What of set theory more specifically? If Quine rejects Carnap’s princi-
ple of tolerance, on what grounds can he urge tolerance in set theory?
In his  “Carnap and Logical Truth,” Quine directly addresses set
theory in his arguments against Carnap. Here, Quine says that set theory
might be accurately described as being true by linguistic convention, sug-
gesting again that his view is much the same as Carnap’s. But we see
Quine immediately emphasizing that he treats set theory as interpreted
mathematics and so as being very much about sets:
In set theory we discourse about certain immaterial entities, real or erro-
neously alleged, viz., sets, or classes. And it is in the effort to make up our
minds about genuine truth and falsity of sentences about these objects that
we find ourselves engaged in something very like convention in an ordinary
non-metaphorical sense of the word. We find ourselves making deliberate
choices and setting them forth unaccompanied by any attempt at justification
other than in terms of elegance and convenience. These adoptions, called
postulates, and their logical consequences …, are true until further notice.
Here, we see already how Quine’s position differs from Carnap’s. Recall
that, in accordance with his principle of tolerance, Carnap thought we
should have the complete freedom to develop a variety of linguistic frame-
works and that we should do so in the pursuit of scientific progress.
Furthermore, Carnap thought that his linguistic frameworks freed him from


As I remarked above, whatever tolerance Quine urges in set theory is one that he will urge for
science as a whole.

W. V. Quine, “Carnap and Logical Truth,” in The Ways of Paradox, p. . In his
“Autobiography,” Carnap say that the principle of tolerance might be more accurately described
as the “principle of conventionality of language forms” (p. ).

Quine, “Carnap and Logical Truth,” p. .
 Quine, New Foundations, and the Philosophy of Set Theory
the metaphysics of the past. In adopting a linguistic framework, we were
only adopting a way of speaking, not committing ourselves to an ontology
in any robust sense. Quine does not share such worries. By focusing on
interpreted theories, he immediately shows his willingness to engage in
ontology. Furthermore, our conventional decisions to treat certain claims
about these entities as true show that these conventions are not prior to our
theorizing about sets; this is not a prior decision to adopt a particular lan-
guage form. Quine’s conventions are a part of the very science of set theory.
Quine goes on to distinguish two types of postulation in which we
might engage here: discursive and legislative. The former is merely the
selecting of some truths as postulates from among the existing body of
truths concerning the relevant entities. This sort of postulation just
imposes some ordering of the truths belonging to a given theory. We
might think of this as what goes on with Euclidean geometry. We have
the truths, and then we select some of them as the postulates from which
the others may be derived. The latter sort of postulation is what we have
here in set theory. This type of postulation takes some claim about the
entities concerned – in this case, sets – and declares our postulate to be
true of them. In this case, postulation is the source of truth. Whereas
Carnap took this latter kind of postulation to mark a significant trait in
distinguishing external from internal questions, Quine does not.
Postulation, whether legislative or discursive, refers simply to that particu-
lar act of postulation and has no enduring consequence. “So conceived,”
he explains, “conventionality is a passing trait, significant at the moving
front of science but useless in classifying the sentences behind the lines. It
is a trait of events and not of sentences.” Even if we could determine
which sentences had originally been adopted as true by legislative postula-
tion so that we might forever identify them as true by convention, this
would be of no help, Quine explains. This would be mere historical con-
jecture, contributing little to our current epistemological understanding of
our scientific theories: “Legislative postulation contributes truths which
become integral to the corpus of truths; the artificiality of their origin does
not linger as a localized quality, but suffuses the corpus.” Once such leg-
islative postulates have been fully integrated into our theory, singling them
out at some later date would be nothing more than discursive postulation.
Still, it might seem as though there is a difference with set theory in
that its legislative postulation never fades, and thus set theory remains


Ibid., p. .

Ibid., pp. –.
 Clarifying Our Conceptual Scheme 
true by linguistic convention in a way that other sciences do not.
However, Quine is very well aware of this point. Indeed, he considers set
theory because of its very explicit and continuing appeals to convention.
This does not run contrary to any of what he says about convention
being a passing trait. Rather, it illuminates how exactly Quine does view
that state of set theory as a science and why he may appear to be much
more openly tolerant on this issue than, say, in his considerations of ele-
mentary logic. What distinguishes set theory for Quine is not so much
its appeals to convention alone but rather what these appeals indicate:
that set theory is at the moving front of science. He takes set theory to
be a largely unsettled science and as such it remains caught up in legisla-
tive postulation. This view would have been fairly uncontroversial from
the discovery of the paradoxes through at least . In coming to terms
with the paradoxes, set theory had yet to return to a conception of set
that was as natural as the original and, unfortunately, inconsistent one.
Given this state, research in set theory proceeded in much in the way
that Quine does it. It was experimental and exploratory; assorted hypoth-
eses and axiom systems were put forward with an eye toward their var-
ious and conflicting consequences, and fundamental questions, such as
the status of the axiom of choice and the continuum hypothesis,
remained open. One of these axiomatizations might “become the com-
mon sense of another generation.”
Here, we might usefully contrast Quine’s seemingly less tolerant attitude
with regard to elementary logic, that is, classical first-order quantification
theory. Unlike set theory, elementary logic is largely a settled science,
fully integrated into our current best theory. While Quine always remains


Quine is explicit about this view of set theory in his  “The Ways of Paradox” reprinted in
The Ways of Paradox, remarking “that the discovery of antinomy is a crisis in the evolution of
thought. In general set theory the crisis began sixty years ago and is not yet over” (p. ).
Although this view of set theory may have lessened later in his career, I think that this remained
his basic attitude toward set theory throughout. I will return to this point in Chapters  and .
This is also the sense in which Quine’s notion of tolerance extends to all of science. At the mov-
ing front of science, that is, in areas that are largely unsettled, we put forward various hypotheses
in hopes that one will eventually turn out to be best. But again, the positing of various hypotheses
is for Quine part of the internal workings of science, which is to say that this is all part of scienti-
fic theorizing and not prior to it.

Quine, “Whitehead and the Rise of Modern Logic,” p. .

While Quine always kept track of the differences between elementary logic – that is, first-order
quantification theory – and set theory, he typically grouped them together under the single head-
ing of logic, as he does in “Carnap and Logical Truth.” I believe that his later insistence on not
grouping set theory with logic is more a shift of emphasis that came with his growing focus on
epistemological matters than a substantive change in his philosophical outlook, but I will not pur-
sue this matter further here.
 Quine, New Foundations, and the Philosophy of Set Theory
open to the possibility of revising elementary logic, at least in a legalistic
sense, he does not actively engage in developing new systems of logic,
very much in contrast with his work in set theory. His grounds for this are
the same that he applies in all areas of science: He thinks that no other pur-
ported logic brings with it, the familiarity, elegance, simplicity, and clarity
of classical first-order quantification theory. Contrary to Carnap, there is
no need to be tolerant with regard to logic. From Quine’s perspective, to
continue to pursue alternatives as genuine logic would itself be an unscien-
tific endeavor, much like continuing to pursue astronomy from the per-
spective of a geocentric universe. We have accepted classical first-order
quantification theory as part of our best current theory and so count it as
true. There is no external question here, only the ongoing progress that is
internal to science as a whole. Set theory, by contrast, was not nearly so
settled a matter.
We might grant that Quine’s position was reasonable – and even
widely accepted – in , but set theory has now matured, moving past
the point at which it was at the moving front of science and becoming a
much more settled matter. Quine himself was very much aware of this
possibility, remarking that “Set theory, currently so caught up in legisla-
tive postulation, may some day gain a norm – even a strain of obvious-
ness, perhaps– and lose all trace of the conventions in its history.”
Perhaps set theory as embodied in the axioms of Zermelo-Fraenkel set
theory with the axiom of choice (ZFC) has now reached this stage, mak-
ing Quine’s position obsolete, dogmatic, and unscientific. We could
argue that, but we should note that set theory still faces many open ques-
tions. It was a great achievement to show that both the axiom of choice
and the continuum hypothesis are independent of the Zermelo-Fraenkel
(ZF) axioms. However, we might still want a set theory that settles these


For a fairly late statement of this view, see Quine’s “Reply to Jules Vuillemen,” in The Philosophy
of W. V. Quine, eds. Hahn and Schilpp, expanded edn, pp. –.

For an example of this attitude, see W. V. Quine, Philosophy of Logic, nd edn (Cambridge, MA:
Harvard University Press, ), p. .

Quine was well aware that there may have been reasons to reconsider logic in light of quantum
mechanics, but on the whole, it seems that such a drastic revision has not maintained much of a
hold on the philosophical imagination. Quine also has some sympathy with the constructive
methods of the intuitionists (Philosophy of Logic, p. ). It is surely an interesting mathematical
endeavor to see what might still be proved by using only their more restrictive means, but this is
not enough to recommend it as a wholesale replacement for our familiar logic. I take it that
Quine would also not object to the exploration of alternative “logics” as abstract algebras, as
many theories of pure mathematics are pursued with no immediate hope of application outside
of themselves, whether in other areas of mathematics or in the natural sciences.

Quine, “Carnap and Logical Truth,” p. .
 Clarifying Our Conceptual Scheme 
questions one way or another. Furthermore, Quine’s particular inter-
est in the “big sets” also escapes investigation by way of ZF. Finally,
there are the non-well-founded sets, such as those that allow for self-
membership, which are also completely left out of ZF, at least when the
axiom of foundation is taken as one of its axioms, as it usually is. I
will address all of these topics in Chapters  and , which I hope will
show that Quine’s particular approach to set theory as explication is still
very relevant and far from an unscientific dogma.


Quine remarks in a fairly late essay that he could see choice as being plausibly accepted as true
but that he has very little in the way of intuitions for or against the continuum hypothesis. On
this point, see his “Kurt Gödel,” reprinted in his Theories and Things, pp. –.

There was long an interest in such sets in the history of set theory – or at least in how we might
talk of collections of absolutely everything in light of the thought that such collections lead to
contradictions; on this point, see Hallett, Cantorian Set Theory and Limitation of Size, pp. –
. That they do not feature in much mainstream research in set theory seems to be the result of
successful propaganda in favor of ZF rather than for compelling mathematical reasons. In fact,
these sets have been found to feature in interesting ways in mathematics. See, for example,
Saunders MacClane, “Locally Small Categories and the Foundations of Set Theory,” in
Infinitistic Methods: Proceedings of the Symposium on Foundations of Mathematics (New York, NY:
Pergamon Press, ), pp. –; and Saunders MacClane, “Categorical Algebra and Set-
Theoretic Foundations,” Proceedings of the Symposium of Pure Mathematics, AMS XIII, Pt. I,
pp. –.

Here, too, interesting and important mathematical research has emerged in recent years.
 PART III

New Foundations and

the Philosophy of Set Theory
 CHAPTER 

The Iterative Conception and Set Theory

The previous five chapters were largely concerned with historical and
expositional issues related to set theory, in particular with the set theory
and philosophy of W. V. Quine. In this chapter, I will continue with
such themes, but here I intend to bring these previous considerations to
bear on some more general issues in the philosophy of set theory. In par-
ticular, I will argue against the iterative conception of set, so often taken
as exemplified by Zermelo-Fraenkel set theory (ZF), as the single correct
version of set theory. Given the variety of set theories discussed in pre-
vious chapters, it might seem surprising that there should be a single cor-
rect version of set. Since the late s, however, set theorists have
tended to give the iterative conception of set, as expressed by ZF, privi-
leged status, as if it captures something of the essence of set, if you will.
There are various reasons why ZF has gained this status; perhaps fore-
most among them has been George Boolos’s excellent exposition of the
iterative conception in his  essay “The Iterative Conception of
Set.” In this paper, Boolos not only makes this conception accessible to
philosophers and mathematicians alike but also does it in such a compel-
ling manner that it is hard not to believe that there is something truly
significant about the notion of set in the iterative conception.
However, I will deny that the iterative conception is the only viable
notion of set. In fact, this recent favoritism for a single conception of set

George Boolos, “The Iterative Conception of Set,” in Logic, Logic, and Logic, ed. Richard Jeffrey,
with Introductions and Afterward by John P. Burgess (Cambridge, MA: Harvard University Press,
), pp. –.

It seems to me that there is something of the last vestige of the a priori in treating ZF in this way.
In a sense, it allows philosophers to hold onto the idea that mathematics and its truths are of a
unique and special variety among all the sciences, not subject to the sort of pragmatic theory-
building constructions that philosophers seem more willing to admit for the traditionally empirical
sciences. Indeed, Hao Wang talks of the intrinsic necessity of set theory depending on the iterative
model; see Hao Wang, “The Concept of Set,” in Philosophy of Mathematics: Selected Readings, eds.
Paul Benacerraf and Hilary Putnam, nd edn, (New York, NY: Cambridge University Press,
), p. .


 Quine, New Foundations, and the Philosophy of Set Theory
is, as perhaps already suspected, the anomaly in the history of set theory.
As we have seen, the development of set theory after the paradoxes as
well as before them has largely been a matter of competing intuitions
about sets. Recall that Georg Cantor, though operating only with an
informal conception of set, perhaps escaped the paradoxes because his
prior theological views barred him from allowing what he deemed abso-
lutely infinite sets, among these the universal set, the set of ordinal num-
bers, and the set of cardinal numbers. Bertrand Russell, by contrast,
struggled to find a consistent notion of set in part because his own intui-
tions led him so naturally to the idea of a universal set. Cantor’s religious
views certainly held no sway for Russell, but he could initially find no
principled way to rule out the problematic sets. Finally, Ernst Zermelo
accepted Cantor’s restrictions but, as we have seen, on largely pragmatic
mathematical grounds. Like many logicians working in this early period
of set theory, Zermelo could see that this was a mathematically interest-
ing theory, and as he saw it, the aim of axiomatization was merely to
capture enough of this theory to ensure its continued interest for mathe-
matical research but to rule out enough that the theory would not give
rise to contradiction. As we have seen, Russell, too, would ultimately
agree, at least to an extent, that this pragmatic criterion was what would
guide the further development of set theory.
In this chapter and Chapter , I will argue that such pragmatic con-
cerns associated with the view that I have identified as set theory as expli-
cation should continue to be the primary factor in developing set theory.
Indeed, I will argue that such pragmatic motivations are much more in
line with both the historical and contemporary development of set the-
ory, and attempting to restrict set theory to ZFC potentially has the
harmful outcome of inhibiting the growth of our mathematical and
scientific knowledge. Therefore set theories such as Quine’s New
Foundations (NF) cannot and should not be ruled out on the grounds
that they somehow stray too far from what was originally intended in the
notion of set. In fact, what I hope to have brought out already and will
continue to bring out here is that the idea of a single intuitive notion of
set, especially as the iterative notion, is largely a myth of set theory’s
founding. I do not want to deny that there is some body of theory that


Joseph Ullian has emphasized that Quine’s exploratory and experimental approach to set theory is
one that encourages the growth of knowledge rather than stifling it. See his “Quine and the Field
of Mathematical Logic,” in The Philosophy of W. V. Quine, eds. Edwin Hahn and Paul A. Schilpp
(Chicago, IL: Open Court, ), p. .
 The Iterative Conception and Set Theory 
we can pick out as set theory. Categorization is certainly a useful prac-
tice. But to the extent to which we can offer criteria for picking out
what counts as set theory, I will argue that NF, too, meets such criteria
and furthers our understanding of sets in general.

. Boolos’s Exposition

To begin this chapter, let us first turn to George Boolos as perhaps the
most important proponent of the iterative conception of set, owing, in
no small degree, to the clarity of his exposition of this conception.
Boolos begins his paper by examining Cantor’s proposed definitions of a
set. Recall that Cantor defined a set as “any collection into a whole … of
definite and separate objects … of our intuition or thought,” or as “every
aggregate of determinate elements which can be united into a whole by
some law.” While Boolos rightly observes that these gesturings at the
definition of set are variously unclear, he also states,
But it cannot be denied that Cantor’s definitions could be used by a per-
son to identify and gain some understanding of the sort of object of which


As Quine himself remarked in “Necessary Truth” in The Ways of Paradox and Other Essays, rev. and
enlarged edn (Cambridge, MA: Harvard University Press, ), p. , “Boundaries between disci-
plines are useful for deans and librarians, but let us not overestimate them – the boundaries. When
we abstract from them, we see all of science – physics, biology, economics, mathematics, logic, and
the rest – as single sprawling system, loosely connected in some portions but disconnected nowhere.”

Boolos gave voice to a view of sets that had been around in the literature but was not widely
known among philosophers or given much prominence among set theorists; see Boolos, “The
Iterative Conception of Set,” p. , fn. . Joseph Shoenfield sketched the view before Boolos,
though in less detail, in his Mathematical Logic (Natick, MA: A. K. Peters, ), pp. –, and
again later in his “Axioms of Set Theory,” in Handbook of Mathematical Logic, ed. Jon Barwise
(New York, NY: North-Holland, ), pp. –. After Boolos, Dana Scott also gave a rather
detailed account of the conception in his “Axiomatizing Set Theory,” in Axiomatic Set Theory,
vol. II, ed. Thomas Jech (Providence, RI: American Mathematical Society, ), pp. –.
Kurt Gödel gives some early expression to this idea already in his  “What Is Cantor’s
Continuum Problem?” in Philosophy of Mathematics: Selected Readings, eds. Paul Benacerraf and
Hilary Putnam, nd edn (New York, NY: Cambridge University Press, ), pp. –. Here,
Gödel distinguishes the iterative conception from the conception of sets as extensions of predicates,
claiming that “the perfectly ‘naïve’ and uncritical working with this concept of set [the iterative]
has so far proved entirely self-consistent” (pp. –). This might be interestingly contrasted with
the attitude he expressed just seven years early, when he claimed that the axiom of foundation,
which says there are no infinitely descending epsilon chains and hence no self-membered sets, is
assumed only on pragmatic grounds of simplifying the work at hand. See his “The Consistency of
the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set
Theory,” in Kurt Godel: Collected Works, vol. II, eds. Solomon Feferman et al. (New York, NY:
Oxford University Press, ), p. . As I stated in the introduction to this chapter, it was largely
following Boolos’s masterly exposition that this conception rose to the fore as the conception
of set.

Cantor, Contributions, p. ; Foundations, p. , n. .
 Quine, New Foundations, and the Philosophy of Set Theory
Cantor wished to treat. Moreover, they do suggest – although it must be
conceded, only very faintly – two important characteristics of sets: that a
set is “determined” by its elements in the sense that sets with exactly the
same elements are identical, and that, in a sense, the clarification of which
is one of the principal objects of the theory whose rationale we shall give,
the elements of a set are “prior to” it.
To illustrate what Cantor might have meant by these hints about what
sets are, Boolos first sketches out what has become known as naïve set
theory. We have seen this conception of a set many times previously.
This is the idea that sets are determined by predicates or are the exten-
sions of predicates. Recall that any given monadic predicate is either true
of an object or false of it. We call the collection of things of which the
predicate is true its extension. It is in this sense that a set can be thought
of as the extension of a predicate. The idea is very intuitive and, as
Boolos notes, seems to make sense of Cantor’s remark that sets are col-
lections of definite elements united by some law. But as we saw in pre-
vious chapters, despite its intuitiveness, the naïve conception is also
inconsistent, as shown by the predicate “is non-self membered,” which
leads directly to Russell’s paradox. Thus the naïve conception of set must
be given up as making sense of Cantor’s proposed definitions of set.
“Faced with the inconsistency of naïve set theory,” Boolos suggests, then,
one might come to believe that any decision to adopt a system of axioms
about set would be arbitrary in that no explanation could be given why
the particular system adopted had any greater claim to describe what we
conceive sets and the membership relation to be like than some other sys-
tem, perhaps incompatible with the one chosen. One might think that no
answer could be given to the question: why adopt this particular system
rather than that or this other one? One might suppose that any apparently
consistent theory of sets would have to be unnatural in some way or frag-
mentary, and that, if consistent, its consistency would be due to certain
provisions that were laid down for the express purpose of avoiding the
paradoxes that show naïve set theory inconsistent, but that lack any inde-
pendent motivation [Boolos’s italics].
I quote this passage at length because it illustrates especially well the
emerging debate between proponents of the iterative conception and a
more pluralistic and experimental understanding of set theory viewed
that was developing in response to the paradoxes. The picture that

Boolos, “The Iterative Conception of Set,” p. .

Ibid., p. –.

Ibid., p. .
 The Iterative Conception and Set Theory 
Boolos sketches here can clearly be identified with certain views of
Quine but I think also with Zermelo and Russell, even if perhaps to a
lesser extent or less explicitly for the latter two. But despite certain claims
from all of them on the apparent ad hoc nature of axiomatizing set the-
ory in light of the paradoxes, I think that this is not entirely true. There
is certainly a guiding intuition behind various approaches to set theory
presented by Zermelo, Russell, and Quine – namely, they all aim to save
as much as possible of the idea that a set is the extension of a concept.
Indeed, Quine’s approach can be seen as a generalization of what is com-
mon to both Zermelo’s and Russell’s set theories, but let us finish laying
out Boolos’s position before returning to these issues.
Boolos goes on to remark that this artificiality does not have to be the
case, as there is another conception of set, the iterative conception, that
strikes him as entirely natural. In this conception, much as in Russell’s
type theory, the sets are formed at stages, or levels. Unlike in Russell’s the-
ory, the stages are cumulative. A set will be any collection that is formed
at a stage. We begin with individuals – again, as Russell did – and at stage
zero, we form all collections of individuals. If no individuals exist, we
form only the empty set at this stage. This will give us all subsets of indivi-
duals, including, of course, the set of all individuals itself. So we get n
sets at this stage, where n is the number of individuals and may be any
natural number or even infinite. We now repeat this process at stage one,
again for any individuals there may be and for all sets that are formed at
stage zero. This process continues for all stages through that natural num-
bers. This brings us to the stage following stages , , , … , that is, stage
omega, where we form all sets of sets of the previous levels, including a set
of all sets formed at these earlier stages. We then continue as before, form-
ing sets in this way through stages omega plus , omega plus , and so on
to omega plus omega (or omega times ), then to omega plus omega plus
omega (omega times ), and so on to omega times omega, and so on. In
this system, the sets are reformed at each stage later than the stage that it
was originally formed at. To simplify matters, we say that a set is formed
only once, at the stage at which it was originally formed. Boolos also
remarks that ZF generally does not assume individuals, so the sets over
which it quantifies are the pure sets – the sets formed in the absence of
any individuals – thus beginning with the empty set ∅, then forming {∅},
then {∅, {∅}}, and so on through the stages as before.


Ibid., p. .

Ibid., pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
Boolos next axiomatizes this stage theory with the aim of ultimately
showing how the axioms of ZF follow from it. He presents the following
nine axioms with variables “x,” “y,” “z,” … ranging over sets; “r,” “s,”
“t,” ranging over stages; the predicates of “=” and “∈” for equality and
membership, respectively, as well as “E ” for “is earlier than” and “F ” for
“is formed at”:
(I) (∀s)¬sEs, no stage is earlier than itself;
(II) (∀r)(∀s)(∀t)((rEs . sEt) → rEt), the transitivity of “earlier than”;
(III) (∀s)(∀t)(sEt ∨ s = t ∨ tEs), for any two stages, either one comes
before the other or they are equal;
(IV) (∃s)(∀t)(t ≠ s → sEt), there is an earliest stage;
(V) (∀s)(∃t)(sEt . (∀r)(rEt → (rEs ∨ r = s))), each stage is followed
immediately by a stage;
(VI) (∃s)((∃t)tEs . (∀t)(tEs → (∃r)(tEr . rEs))), there is a stage aside
from the first stage that does not immediately follow any stage,
such as the omega stage;
(VII) (∀x)(∃s)(xFs . (∀t)(xFt → t = s)), every set is formed at a unique
stage;
(VIII) (∀x)(∀y)(∀s)(∀t)((y ∈ x . xFs . yFt) → tEs), members of sets are
formed before the set itself; and
(IX) (∀x)(∀x)(∀t)(xFx . tEs → (∃y)(∃r)(y ∈ x . yFr . (t = r ∨ tEr), if a
set is formed at a stage, then at or after any earlier stage, at least
one of its members has also been formed.
The first five axioms here are meant to govern the stages, while the last four
describe at which stages sets and their members are formed. Boolos adds
two additional axiom schemata: one for set specification and one for induc-
tion with regard to sets and stages. The specification axioms are of the form
(∀s)(∃y)(∀x)(x ∈ y ↔ (χ . (∃t)(tEs . xFt))), where χ is a formula of our
language with no occurrence of “y” free.
These axioms state that for any stage, there is a set formed whose mem-
bers are just those sets that were formed at earlier stages of which the for-
mula χ is true. The induction axioms are of the form
(∀s)(∀t)(tEs → (∀x)(xFt → θ)) → (∀x)(xFs → χ)) → (∀s)(∀x)(xFs → χ),
where χ is a formula of our language containing no occurrences of
“t” and θ is like χ except that it contains free occurrences of “t”
wherever θ contained occurrences of “s.”
Intuitively, these axioms say that if a stage is covered by a predicate pro-
vided that all earlier stages are covered by it, then all stages are covered
 The Iterative Conception and Set Theory 
by that predicate. Boolos defines “a stage being covered by a predicate”
as “the predicate holding of all sets formed at that stage.” The point of
all this axiomatizing is to show next that we can derive the usual axioms
of Zermelo set theory (Z) from this account of the stage theory. In this
sense, Boolos claims that the iterative conception is a natural account of
sets on a par with the naïve conception.
Recall that the axioms of Z are the empty set axiom, pairing, union,
power set, infinity, separation, and extensionality. Boolos also includes
regularity, or foundation, among these axioms. Extensionality, he says,
“has a special status.” We will return shortly to each of these axioms in
turn. Also, to simplify matters, as is usual, Boolos assumes that there are
no individuals, only sets. We saw some version of the Zermelo axioms in
Chapter , but to reiterate, Boolos states them as follows:
() Empty set: (∃y)(∀x) ¬x ∈ y, there is a set with no members;
() Pairing: (∀z)(∀w)(∃y)(∀x)(x ∈ y ↔ (x = z ∨ x = w)), for any sets z
and w, there is a set with z and w as its only members;
() Union: (∀z)(∃y)(∀x)(x ∈ y ↔ (∃w)(x ∈ w . w ∈ z)), for any set z,
there is a set with just the members of z as its members;
() Power set: (∀z)(∃y)(∀x)(x ∈ y ↔ (∀x)(w ∈ x → w ∈ z)), for any
set z, there is a set with just the subsets of z as members;
() Infinity: (∃y)((∃x)(x ∈ y . (∀z) ¬ z ∈ x) . (∀x)(x ∈ y → (∃z)(z ∈ y .
(∀w)(w ∈ z ↔ (w ∈ x ∨ w = x))))), calling z a successor of x if the
members of z are only the members of x and x itself, there is a set
containing an empty set and the successor of any set it contains;
() Separation: (∀z)(∃y)(∀x)(x ∈ y ↔ (x ∈ z . φ)), where φ is a formula
in which y does not occur free. For any set z, there is a set y of just
those members of z for which φ holds;
() Regularity, or foundation: (∃x) φ → (∃x)(φ . (∀y)(y ∈ x→ ¬ ψ)),
where y does not occur free in φ and ψ is like φ except that it
contains an occurrence of y wherever φ has a free occurrence of x. If
there is a set x for which φ holds, then there is a set x for which φ
holds but containing members y for which this formula does not hold.
Let us present just one example from Boolos of how the axioms of Z fol-
low from the stage theory. Take, for example, the empty set axiom
“(∃y)(∀x)(¬x ∈ y),” which states that there is a set with no members.


Ibid., pp. –.

Ibid., p. .
 Quine, New Foundations, and the Philosophy of Set Theory
If we let χ be “x = x,” this yields “(∀s)(∃y)(∀x)(x ∈ y ↔ (x = x . (∃t)(tEs .
xFt))),” which is a specification axiom stating that for any stage, there is a
set of all sets formed at some earlier stage. Since there is an earliest stage,
there is a stage that has no sets formed before it. Hence there is a set with
no members, so the empty set axiom follows from the stage theory. The
other axioms of Z follow similarly, although replacement and choice do
not. Adding these two axioms gives us the full set theory ZFC, that is,
Zermelo-Fraenkel set theory with the axiom of choice.

. The Iterative Conception As Set Theory

There is no doubt something very appealing about the view of sets that
we have just sketched, but in this section, I want to take a more critical
attitude toward the iterative conception by looking at the reasons Boolos
gives in its favor as well by trying to bring out some of its less intuitive
features. Following his remarks on the supposed ad hoc nature of adopt-
ing any set theory in light of the paradoxes, Boolos remarks that the
iterative conception “often strikes people as entirely natural, free from
artificiality, not at all ad hoc, and one they might perhaps have formu-
lated themselves.” This may be true, but so far, this only makes it a
competitor with other conceptions of set. As we saw in Chapter , it
might have seemed to Cantor entirely natural, for example, to rule out a
universal set, given his background metaphysical views, but as we saw in
that chapter, Russell found the opposite view so appealing that he could
not see his way to fully accepting his own early proposal of type theory.
Even with types in its initial Principles of Mathematics formulation,
Russell thought that there must be a class of all terms, or, more generally,
objects, and this would regenerate the paradoxes. Similarly, Zermelo,
for whom the set theory Boolos describes is named, put forward his
axioms with a largely pragmatic justification aimed at preserving as
much as was mathematically interesting from Cantor’s original theory.

Ibid., pp. –.

Ibid., p. .

Russell, Principles, pp. –. A similar situation arises also for propositions (Principles, pp. –).
For more detail, see Hylton, Russell, Idealism, and the Emergence of Analytic Philosophy, pp. –.

Wang calls such approaches to restoring consistency to set theory “bankruptcy approaches” (see
“The Concept of Set,” pp. –). He contrasts this with misunderstanding approaches, which
search for some misunderstanding in our original understanding of a set. He claims Cantor as
part of this tradition. I am not so sure. It seems to me that Cantor might never have understood
set theory to embody something paradoxical. It is unclear whether he introduced his distinction
between consistent and inconsistent totalities in reaction to paradox or whether this distinction
was always in place, at least implicitly.
 The Iterative Conception and Set Theory 
My point here is just to signal that intuitions alone, even the intuitions
of set theory’s founders themselves, will not settle questions about which
set theory we are to accept. Boolos, I think, ultimately accepts this.
Indeed, he continues, observing of the iterative conception,
It is, perhaps, no more natural a conception than the naïve conception,
and certainly not quite so simple to describe. On the other hand, it is, as
far as we know, consistent: not only are the sets whose existence would
lead to contradiction not assumed to exist in the axioms of the theories
that express the iterative conception, but the more than fifty years experi-
ence that practicing set theorists have had with this conception have
yielded a good understanding of what can and what cannot be proved in
these theories, and at present there just is no suspicion at all that they are
inconsistent.
So in addition to its naturalness, the iterative conception appears to be
consistent. But this seems to hold for any serious alternatives to the itera-
tive conception. Certainly, type theory also appears to be consistent, as
does NF. And while perhaps neither theory has received the scrutiny of
the iterative conception as embodied in ZF, there does exist a vast
amount of research on each, similar to the sort of research that set theor-
ists have carried out on ZF. So consistency alone does not uniquely pick
out the iterative conception from among other set theories, particularly
in the experimental form that Boolos suggests here: that we think ZF
consistent because in working with it, we have yet to find a contradic-
tion. We should also flag another theme here that will continue to arise.
Boolos remarks that “the sets whose existence would lead to contradic-
tion [are] not assumed to exist in the axioms of the theories that express
the iterative conception,” but this raises another question. Are other sets,
perhaps of either mathematical or philosophical interest, banned from
these theories? The obvious possibility is that that none of them yield a
universal set, in contrast to NF. This is an issue to which we will return
shortly in more detail, considering NF more directly.


Boolos, “The Iterative Conception of Set,” p. .

See, for example, Gerhard Gentzen, “The Consistency of the Simple Theory of Types,” in The
Collected Papers of Gerhard Gentzen, ed. M. E. Szabo (Amsterdam: North-Holland, ),
pp. –. Gentzen does not include infinity as one of the axioms of the simple theory of types,
though he does include choice.

We saw earlier Rosser’s account that the usual sorts of contradictions do not arise in NF. Thomas
Forster has conveyed to me in conversation that even a contradiction in NF would be exciting,
since it would have to arise in wholly unexpected ways. Rosser, J. Barkley, “On the Consistency
of Quine’s New Foundations for Mathematical Logic.” The Journal of Symbolic Logic (March
), pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
At this point, Boolos himself turns directly to NF (and its extension,
ML). He remarks first that ZF is the standard first-order theory for
expressing the iterative conception (also observing that the subsystem Z
also embodies this idea as well as the extended systems Von Neumann-
Bernays-Gödel (NBG) and Morse-Kelley (MK)). But he says, of theories
that are proposed to be incompatible with ZF, by which he means NF
and ML, that
[t]hese theories appear to lack a motivation that is independent of the
paradoxes in the following sense: they are not, as Russell has written,
“such as even the cleverest logician would have thought of if he had not
known of the contradictions.” A final and satisfying resolution of the set-
theoretical paradoxes cannot by embodied in a theory that blocks their
derivation by artificial technical restrictions on the set of axioms that are
imposed only because [Boolos’s italics] paradox would otherwise ensue;
these other theories survive only through such artificial devices. ZF alone
(together with its extensions and subsystems) is not only a consistent (appar-
ently) but also an independently motivated theory of sets: there is so to speak,
a “thought behind it” about the nature of sets which might have been put
forth even if, impossibly, naïve set theory had been consistent [my italics].
The thought, moreover, can be described in a rough, but informative way
without first stating the theory the thought is behind.
At least in part, Quine would accept Boolos’s point, and so would
Zermelo – and even Russell in certain moods. This was the point of
Quine’s remark that in light of the contradictions “commonsense is
bankrupt.” Any resolution of the paradoxes will have some degree of
unnaturalness about it. But this is not to say that set theories such as
type theory or NF are wholly unmotivated. The development of both
types and NF is guided by capturing as much of comprehension in its
original form as possible as the basis for set existence. ZF’s separation is
perhaps just that much farther from the original insight of set existence
as a set being the extension of a predicate. Furthermore, type theory has
very much the same sort of hierarchical structure that the iterative
notion, as found in ZF, does, and NF takes its guiding thought from
types and Russell’s use of typical ambiguity in expositing his system.
Indeed, we can even understand NF as taking this idea of a hierarchy,
found in both type theory and ZF, and abstracting away from any actual
layering of the universe. We can then view the insight of both types and
ZF being not that the universe actually comes in layers but rather that

Boolos, “The Iterative Conception of Set,” p. . The Russell quote is from My Philosophical
Development , p. .
 The Iterative Conception and Set Theory 
the subscripting of variables can provide a syntactic test for set existence
without the extra, perhaps metaphysical, claim that the universe comes
in a prearranged hierarchy. Both the idea of sets as the extensions of
predicates and the ideas of sets being layered in a hierarchy point to ZF
as not being the only independently motivated set theory with an easily
describable thought behind it.
But Boolos thinks that the iterative conception brings out a very parti-
cular motivating thought that NF does not have. He next explains,
Whatever tenuous hold on the conceptions of set and member were given one
by Cantor’s definitions of “set” and one’s ordinary understanding of “ele-
ment,” “set,” “collection,” etc. is altogether lost if one is to suppose that some
sets are members of themselves. The idea is paradoxical not in the sense that
it is contradictory to suppose that some set is a member of itself, for, after all,
“(∃x)(Sx . x ∈ x)” is obviously consistent, but that if one understands “∈”
as meaning “is a member of,” it is very, very peculiar to suppose it true. For
when one is told that a set is a collection into a whole of definite elements of
our thought, one thinks: Here are some things. Now we bind them up into a
whole. We don’t suppose that what we come up with after combining some
elements into a whole could have been one of the very things we combined
(not, at least, if we are combining two or more elements) [Boolos’s italics].
NF certainly does allow for self-membered sets, for example, in its universal
set V given by the instance of the comprehension schema “(∃y)(∀x)(x ∈ y ↔
x = x).” Then V is certainly a member of V because it is self-identical.
This may very well seem counterintuitive in thinking about sets along the
lines sketched by Boolos. As he presents it here, set theory sounds like a
theory of collections of very ordinary physical objects: Here are some
things, and we collect them together. This sounds fine when we think

In , Russell also proposed the zigzag theory as a possible solution to the paradoxes stating,
“In the zigzag theory, we start from the suggestion that propositional functions determine classes
when they are fairly simple, and only fail to do so when they are complicated and recondite.”
However, he could not find criteria for determining when a propositional function became too
complicated. See his “On Some Difficulties in the Theory of Transfinite Numbers and Order
Types,” pp. –. Michael Potter observes that Quine’s NF perhaps makes sense of Russell’s
zigzag theory and also observes this in his Set Theory and Its Philosophy (New York, NY: Oxford
University Press, ), p. .

Boolos, “The Iterative Conception of Set,” pp. –; Boolos’s italics.

Wang also talks this way in discussing the collections that make up set theory. He uses the exam-
ple of two tables in a room and the idea that by looking at them, pointing at them, or thinking
about them in the right sort of way, we might view them separately or as a unity; see “The
Concept of Set,” p. . Charles Parsons, in an attempt to get away from this idealistic view of
how sets come to be, instead puts forth an account in terms of potentiality and actuality in his
“What Is the Iterative Conception of Set?” in Philosophy of Mathematics, pp. –. While
Parsons’s account is perhaps more in line with the Platonist tendencies of most set theorists, it
depends upon our making rigorous sense of the controversial modal notions.
 Quine, New Foundations, and the Philosophy of Set Theory
about, say, rocks or paper clips, but this brings us back to an issue that we
have faced before: How intuitive is set theory? In its most intuitive version,
it was, of course, inconsistent. But in the gloss that Boolos gives it here, it
would hardly make sense of abstract collections of numbers or, say, of sets
themselves: Here are some numbers. Now we bind them up into a whole.
This is only a metaphor – perhaps a useful one, but a metaphor does not
make rigorous sense of the concept of set theory as a mathematical theory of
collections. I am at a loss, as I think we should be, when confronted by
this situation along the same lines of our rock or paper clip collections.
However, the abstract theory as embodied in the idea of a set as the exten-
sion of predicate makes reasonably clear the sense in which sets are collec-
tions of objects. This idea also seems to make sense of Cantor’s remark
about combining into a whole by a law. I can collect rocks, paper clips,
numbers, and even sets according to whether or not a predicate is true of
them. In its most general form, this idea led to contradiction. So intuitions
are to be used with great caution. But this idea does guide us as to how set
theory might use the notions of set and membership and the sense in which
a set can be a member of itself. What it means for an object to be a member
of a set is just for a certain predicate to be true of that object. This might not
be our most ordinary understanding of set and membership, but it is one of
the guiding thoughts that set theory began with.
Boolos’s metaphor seems to fail in other ways as well. What made set
theory so important was its ability to make sense of the infinite. This is
one of the minimal requirements of what any consistent set theory should
do. How would our collecting together an infinite number of objects be
made sense of according to our ability to bind objects together? Certainly,
we cannot do this by running around grabbing up an infinite number of
objects. Perhaps the metaphor helps if we think about a finite number of
objects and then we can always just keeping adding one more. (I am not
sure that it does help, especially as the idea that follows is that of the
potential infinite, not the actual infinite as Cantor intended.) Even if this
does seem to get us farther along in understanding infinite collections, it
seems to fail again when we start talking about uncountable infinities and
infinities of ever increasing size. None of this is meant to be definitive
against Boolos, but it is to urge strong caution in our appeal to intuitions
and metaphors when discussing set theory. The extent to which we have

I am not suggesting that the extension of a predicate view is the only way in which we might
make sense of set theory. I only urge it as a contender, and one that perhaps fares better when it
comes to making sense of collections of objects, abstract or otherwise, in the set-theoretic sense or
in the sense of self-membership.
 The Iterative Conception and Set Theory 
any reliable intuitions about set theory seems to me very limited, and it
can hardly be a deciding factor in determining which set theories are
worthy of our investigations as areas of research. Indeed, much of what we
find interesting about set theory runs quite contrary to our commonsense
view of the world until we begin learning set theory.
Both of these issues raise a further question about the relationship of
time to the iterative conception of set. According to the iterative concep-
tion, as Boolos describes it, sets are formed at stages. So we have not only
the question discussed earlier of how we collect objects into sets but also
the question of when. It seems counterintuitive to think of sets being cre-
ated at particular points in time, either past or future. Set theorists tend to
work on the assumption that all the sets are already available to them, that
they do not need to wait for some perhaps very large sets to be created at a
future time (and since we can always “keep going” in the iterative hierar-
chy, it would seem that there are always sets not yet formed). Nor do they
wonder at which point in the past a particular set came into being. Both
questions sound absurd (although the question of when a particular set
was discovered or the idea that more sets might be discovered in the future
are not similarly absurd topics). Wang tries to put such questions aside by
claiming that set formation, as the iterative conception describes it, is an
idealization of human capacities, in terms of both our collecting abilities
and time. But we have to ask again: How ideal is this idealization?
Again, it seems to rest on intuitions of which set theory was supposed to
make sense in the first place. As Parsons observes, the kinds of intuitions
that Wang would need to make sense of the iterative conception’s idea
of set formation would go not only beyond an intuition of the potential
infinite – our ability to always perform a further step – but beyond limita-
tions imposed by space–time structure itself.
But perhaps Boolos admits that a battle among intuitions will not be
resolved. As he concludes,
There does not seem to be any argument that is guaranteed to persuade
someone who really does not see the peculiarity of a set’s belonging to
itself, or to one of its members, etc., that these states of affairs are peculiar.

Parsons, among others, has focused on this difficulty for the iterative conception. Trying to cope
with this issue led him to his account of set existence in terms of potentiality and actuality. See
his “What Is the Iterative Conception of Set?”

Wang, “The Concept of Set,” pp. –.

Parsons, “What Is the Iterative Conception?” p. . NF seems to fare better on the issues of set
formation raised in this paragraph and on previous pages, since it allows us to say that the sets are
already there. NF’s comprehension principle then allows us to specify certain sets, but there is no
talk of the sets being created by us or anything else.
 Quine, New Foundations, and the Philosophy of Set Theory
But it is in part the sense of their oddity that has led set-theorists to favor
conceptions of set such as the iterative conception, according to which
what they find odd does not occur.
Again, we might say that “oddity” is a relative term, and much of what any
set theorist would say to the uninitiated would sound quite odd indeed!

. The Iterative Conception and Zermelo’s Axioms

I want now to consider more carefully some of the axioms of Z that
both follow and do not follow from the stage theory. I will begin with
replacement and extensionality, as neither follows from the iterative
conception and this seems quite uncontroversial. However, both repla-
cement and extensionality raise interesting questions about whether the
iterative conception deserves the privileged status it has gained. I will
then turn to regularity, or foundation, which Boolos says does follow
from the stage theory. This will bring us back to one of the issues that
I considered in Section .: that it is too strange to say that a collection
can be a member of itself. The axiom of choice will be the last of the
axioms I wish to examine. It, too, does not follow from the iterative
conception, but as this leads to much broader issues concerning NF and
set theory more generally, I will postpone discussion of the axiom of
choice until Chapter .
Let us begin with replacement, since its absence from the iterative
conception will be the least controversial. ZF, as Boolos describes it, con-
sists of the previous axioms of Z plus the replacement axiom schema:
F is a function → (∀z)(∃y)(∀x)(x ∈ y ↔ (∃w)(w ∈ z . F(w) = x)), if F
is a function, then if its domain is a set, so is its image.
Boolos thinks it possible that we could have allowed an extension of the
stage theory from which replacement would follow such as all instances
of the following:
If each set is correlated in some way with at least one stage, then for any
set z there is a stage s such that for each member w of z, s is later than
some stage with which w is correlated.
But while Boolos thinks that this “is an attractive further thought about
the interrelation of sets and stages … it does seem to us to be a further
thought, and not one that can be said to have been meant in the rough

Boolos, “The Iterative Conception of Set,” p. .
 The Iterative Conception and Set Theory 
description of the iterative conception” [Boolos’s italics]. For example,
it could be that there are exactly ω stages, since nothing in the rough
account of the stage theory rules this out. In such a case, replacement
would not hold generally. On the other hand, replacement has some well-
known desirable consequences for the theory of sets. For example, it allows
us to define a sequence of sets {Rα} with which each stage can be identified
if we let R = ∅; Rα+ = Rα ∪ P(Rα) (the power set of Rα); and Rλ = ∪β>λ
Rβ, where λ is a limit ordinal. Replacement assures us that R is well defined,
s is a stage if (∃α)(s = Rα), x is formed at s if x ⊆ s and x ∉ s, and that s is ear-
lier than t if (∃α)(∃β)(s = Rα and t = Rβ, and α < β). Replacement allows
us to go beyond translating the axioms of the stage theory into the language
of set theory to stronger axioms asserting the existence of stages farther out
than those suggested by the rough account of the stage theory.
Echoing Russell’s inductive justification of reducibility, Boolos then
states,
Although they are not derived from the iterative conception, the reason for
adopting the axioms of replacement is quite simple: they have many desir-
able consequences and (apparently) no undesirable ones. In addition to the-
orems about the iterative conception, the consequences of replacement
include a satisfactory if not ideal [fn. An ideal theory would decide the con-
tinuum hypothesis, at least.] theory of infinite numbers, and a highly desir-
able result that justifies inductive definitions on well-founded relations.
While this remark is perfectly acceptable for a view of set theory carried out
in a pragmatic and experimental spirit, it seems to run somewhat contrary to
Boolos’s declared preference for the iterative conception over all other ver-
sions of set theory. While the iterative conception is perhaps intuitive and
easy to describe, we see here that certain important mathematical features do
not follow from it until we add the replacement axiom. Again, adding the
axiom seems perfectly acceptable if our guiding thought in developing
set theory is to maintain as much as possible of what is mathematically inter-
esting about it in light of the paradoxes. But if set theory is supposed to be


Ibid., pp. –.

Ibid., p. .

Ibid. We will return shortly this idea of an ideal theory in considering the axiom of choice.

Of course, stronger claims have made for favoring certain axioms of set theory. Gödel and Wang
both talk of axioms “forcing themselves upon us,” although it does not seem entirely clear what
this forcing consists of. See Kurt Gödel, “What Is Cantor’s Continuum Problem?,” p. , and
Wang, “The Concept of Set,” p. , respectively.

In particular, the iterative conception, as Boolos describes it so far, without replacement leaves us
with a set theory in which some sets do not have a cardinal number of members. For details, see
W. D. Hart, The Evolution of Logic (New York, NY: Cambridge University Press, ), pp. –
. Such a set theory, to use Hart’s words, “would be crippled.”
 Quine, New Foundations, and the Philosophy of Set Theory
grounded in the iterative conception to the exclusion of all other concep-
tions of set theory, we seem to be caught at an uncomfortable halfway point.
On the one hand, we privilege ZF over all other set theories because it fol-
lows from the iterative conception and so gives expression to this intuitive
notion of set formation. On the other hand, we see that the iterative con-
ception is not strong enough to capture important mathematical features of
set theory. Thus we are left adding axioms in a somewhat ad hoc way so as
to restore the power that was lost in restricting the conception of set to
avoid contradiction. I do not mean to suggest that we should now instead
privilege some other version of set theory – say, NF –, over ZF. I simply
want to point out that the intuitive picture that was supposed to lead us
naturally to prefer ZF only gets us so far, mathematically speaking.
Let us next consider the axiom of regularity, or foundation, which
Boolos says is among the axioms of Z that follow from the iterative concep-
tion. It most certainly does follow, as is demonstrated by Boolos’s proof
sketch. Let φ hold of a set x′. Then x′ is formed at some stage that is not
covered by ¬φ, since ¬φ is false of x′. By an induction axiom, then, there
is an earliest stage s not covered by ¬φ, though all stages earlier than s are
covered by ¬φ. Hence there is an x formed at stage s of which ¬φ does not
hold and therefore of which φ does hold. If y is in x, then y is formed
before stage s, so the stage at which y is formed is covered by ¬φ. Hence
¬φ holds of y, and this is what ¬ψ says. What is tendentious about this
is Boolos’s claim that the regularity axiom is in fact an axiom of Z.
Recall from Chapter  that Zermelo does not assume this axiom as
one of the axioms of his system. Indeed, it was not explicitly proposed as
an axiom until von Neumann’s work of the mid-s, and Zermelo
assumes it on the purely pragmatic grounds of simplifying his set theory
in light of the particular proof he is carrying out. Zermelo himself
was the first to explicitly adopt foundation as a general axiom of set the-
ory but did not do so until . At least as late as , Fraenkel,
Bar-Hillel, and Levy state it merely as optional among axioms for Z.
So while the idea might look intuitive now in light of the iterative


Boolos, “The Iterative Conception of Set,” pp. –.

John von Neumann, “Ueber eine Widerspruchfreiheitsfrage in der axiomatischen Mengenlehre,”
in John von Neuman: Collected Works, vol. I, ed. Abraham H. Taub (New York, NY: Pergamon
Press, ), pp. –.

Ernst Zermelo, “On Boundary Numbers and Domains of Sets: New Investigations in the
Foundations of Set Theory,” in From Kant to Hilbert: A Source Book in the Foundations of
Mathematics, vol. II, ed. William Ewald (Oxford: Clarendon Press, ), pp. –.

Fraenkel, Bar-Hillel, and Levy, Foundations of Set Theory, pp. –. A little later in the work,
they remark that since foundation “is not essential for mathematics, it cannot be regarded as fun-
damental by the traditional axiomatic attitude” (p. ).
 The Iterative Conception and Set Theory 
conception and in light of the paradoxes, it was, oddly enough, not an
intuition about sets that occurred to many at the founding of the theory.
Indeed, while it might not be a further thought about the iterative con-
ception, it does seem a further thought about set theory in general.
What, then, recommends regularity? In short, the axiom says that
there are no self-membered sets or sets with circular membership condi-
tions, that is, no sets such that x ∈ y and y ∈ x. Recall that Boolos
argued that it is too strange to say that a collection can be a member of
itself. But I countered that the metaphor driving this intuition does not
hold up. Boolos offers no justification for it other than that it follows
from the stage theory he sketches. I do not think that this is necessarily
unfair in all cases as a justification, but here, it prejudices our choice of
set theory. While other set theories also express much of what the ZF
axioms do, a set theory that allows for non-well-founded sets would be
automatically ruled out by regularity. If what it is to be a set is to be well
founded, then a set theory that includes non-well-founded sets is ruled
out from the start. Boolos’s claim, then, about ZF alone being an appar-
ently consistent and independently motivated account of sets seems to
stand automatically unchallenged. Starting with the iterative conception,
as Boolos describes it, drives us to accept ZF (including regularity) and
its subsystems and extensions as the set theory.
But there are other motivations for accepting regularity as an axiom of
set theory. Although it seems that Mirimanoff first entertained the idea
of focusing on well-founded sets in , followed by von Neumann in
his  “An Axiomatization of Set Theory,” it was Zermelo who fully
accepted regularity as an axiom in his  “On Boundary Numbers
and Domains of Sets.” Neither von Neumann nor Zermelo claimed
that the universe of sets was actually restricted to well-founded sets.

For another sort of response to Boolos, see Thomas Forster, “The Iterative Conception of Set,”
Review of Symbolic Logic, : (), pp. –. Forster argues here that the iterative conception
is broader than cumulative hierarchy and includes some non-well-founded sets.

Dimitri Mirimanoff, “Les Antinomies de Russell et de Burali-Forti et le Problème Fondamental
de la Théorie des Ensembles,” L’Enseignement Mathématique,  (), pp. –, and his
“Remarques sur la Théorie des Ensembles,” L’Enseignement Mathématique,  (), pp. –
; John von Neumann, “An Axiomatization of Set Theory,” in From Frege to Gödel: A Source
Book in Mathematical Logic, –, ed. Jean van Heijenoort (Cambridge, MA: Harvard
University Press, ), pp. –; Zermelo, “On Boundary Numbers and Domains of Sets,”
pp. –. Details of this history can be found in various locations; see, for example, Akihiro
Kanamori, “The Mathematical Development of Set Theory from Cantor to Cohen,” The Bulletin
of Symbolic Logic : (), pp. –; Potter, Set Theory and Its Philosophy, pp. –; or Adam
Rieger, “Paradox, ZF, and the Axiom of Foundation,” in Vintage Enthusiasms: Essays in Honor of
J. L. Bell, eds. Peter Clark, Michael Hallett and D. DeVidi (Glasgow ePrints Service: http://
eprints.gla.ac.uk/, ), p. .
 Quine, New Foundations, and the Philosophy of Set Theory
Rather, they both introduced the axiom for the practical purpose of
obtaining categoricity results for their axiom systems. Regularity simpli-
fied the universe by eliminating non-well-founded sets, but none of
them saw well-foundedness as essential to the concept of set. As Zermelo
made the point with regard to accepting regularity: “This … axiom,
which excludes all ‘circular’ sets, and all ‘sets that contain themselves’,
and in general all ‘groundless’ sets, has always been satisfied in all practi-
cal applications of set-theory. Thus, for the time being, it presents no
essential restriction to the theory.”
More recent justifications for regularity rest on similar grounds in that
regularity simplifies the set-theoretic universe particularly with regard to
inductive definitions and investigating models of set theory. This attitude
is standard in set theory texts. As is well known, all of the other axioms
of ZF hold, regardless of regularity. Furthermore, all ordinary mathe-
matics takes place within a universe of well-founded sets. So with regard
to ease of use as well as perhaps Occam’s Razor, regularity may seem
quite worthwhile to assume, regardless of whether we think that this is
inherent in the notion of set. Jech makes this point in his standard intro-
duction to set theory:
It should be stressed that, whether or not one accepts the Axiom of
Foundation, makes no difference as far as the development of ordinary
mathematics in set theory is concerned. Natural numbers, integers, real
numbers and functions on them, and even cardinal and ordinal numbers
have been defined, and their properties proved in this book, without any
use of the Axiom of Foundation. As far as they are concerned, it does not
make any difference whether or not there exist any non-well-founded sets.
However, the Axiom of Foundation is very useful in investigations of
models of set theory.
Kanamori goes somewhat further in his history of set theory, remarking
that regularity is in fact what makes set theory its own special branch of
mathematics:
It is nowadays almost banal that Foundation is the one axiom unnecessary
for the recasting of mathematics in set-theoretic terms, but the axiom is
also the salient feature that distinguishes investigations specific to set the-
ory as an autonomous field of mathematics. Indeed, it can be fairly said


Zermelo, “On Boundary Numbers and Domains of Sets,” p. .

See, for example, Karel Hrbacek and Thomas Jech, Introduction to Set Theory, rd rev. and
expanded edn (New York, NY: Marcel Dekker, ), p. , and Kenneth Kunen, Set Theory:
An Introduction to Independence Proofs (New York, NY: North-Holland, ), pp. –.

Hrbacek and Jech, Introduction to Set Theory, p. .
 The Iterative Conception and Set Theory 
that current set theory is at base the study of well-foundedness, the
Cantorian well-ordering doctrines adapted to the Zermelian generative
conception of sets.
However, he is also careful to note that it is “a notable inversion [that]
this iterative conception became a heuristic for motivating the axioms of
set theory generally.”
So there do seem to be some distinctly mathematical reasons for prefer-
ring a set theory of well-founded sets as given by regularity. But is this
attitude toward the well-founded sets an accurate picture of current set-
theoretic research? It seems not. Particularly notable in this context is
Aczel’s work on non-well-founded sets. Since its publication, interest in
non-well-founded set theory has become widespread, with important
applications in computer science. Philosophers, too, have recognized that
such set theories have important applications, particularly in modeling
ordinary language and in coming to grips with the semantic paradoxes.
Many of the current texts on set theory now include sections dedicated to
non-well-founded sets. Furthermore, Kanamori himself has noted that
set theory itself has now become an autonomous subject of mathematical
investigation rather than merely serving as a foundation for all other
mathematics. If we view set theory in this way, it seems that we should be
interested in all sets, not just the well-founded ones. Should we not also
want to broaden our investigations to understanding what the non-well-
founded ones are like? So it seems that non-well-founded sets do have a
place in the mathematics and philosophy surrounding set theory. Indeed,
it is perhaps fair to say that out-of-hand dismissals of such set theories
have impeded the acquisition of mathematical knowledge.


Kanamori, “Mathematical Development of Set Theory,” pp. –.

Peter Aczel, Non-Well-Founded Sets (Stanford, CA: CSLI Publications, ).

The most notable work here is Jon Barwise and John Etchemendy, The Liar: An Essay on Truth
and Circularity (New York, NY: Oxford University Press, ). W. D. Hart has suggested that
we might use set theory with a universal set to give a better account of the universal quantifier, as
this would give a sense in which “all” really does mean all. See his The Evolution of Logic,
pp. –. In a recent collection of essays edited by Agustin Rayo and Gabriel Uzquiano,
Absolute Generality (New York, NY: Cambridge University Press, ), some mention is made
of this, but the idea is quickly dismissed on grounds similar to what Boolos says at the beginning
of his paper on the iterative conception. See in particular Oystein Linnebo, “Sets, Properties, and
Unrestricted Quantification,” in Absolute Generality, pp. –; and Alan Weir, “Is It Too Much
to Ask, to Ask for Everything?,” in Absolute Generality, p. .

See, for example, Hrbacek and Jech, Introduction to Set Theory; Yiannis Moschovakis, Notes on Set
Theory, nd edn (New York, NY: Springer, ); and Devlin, The Joy of Sets. Rieger observes
that the picture on the cover of Devlin’s book is in fact of a non-well-founded set. See his “An
Argument for Finsler-Aczel Set Theory,” Mind : (), p. , fn. .

Kanamori, “Mathematical Development of Set Theory,” p. .
 Quine, New Foundations, and the Philosophy of Set Theory
I want to turn now to the extensionality axiom and the special episte-
mological status that Boolos says that it has, which, like replacement,
does not follow from the iterative conception. Boolos observes that if
someone were to deny any other axiom of ZF, we would be more
inclined to believe the axiom false than if someone were to deny exten-
sionality. The claim “There are distinct sets with the same members”
seems so far from any ordinary conception of set that it would justify
our believing that the asserter of such a statement must have some non-
standard usage of the word “set,” that they are not merely claiming
extensionality to be false. “Because of this difference,” he concludes, “one
might be tempted to call the axiom of extensionality ‘analytic,’ true by
virtue of the meanings of the words contained in it, but not to consider
the other axioms analytic.” Analyticity is, of course, a controversial
notion, and Boolos recognizes this, noting that until we have an account
of how a sentence can be true by virtue of meaning, we should refrain
from classifying extensionality as analytic. Still,
[i]t seems probable, nevertheless, that whatever justification for accepting
the axiom of extensionality there may be, it is more likely to resemble the
justification for accepting most of the classical examples of analytic sen-
tences, such as “all bachelors are unmarried” or “siblings have siblings”
than is the justification for accepting the other axioms of set theory. That
the concepts of set and being a member of obey the axiom of extensionality
is a far more central feature of our use of them than is the fact that they
obey any other axiom. A theory that denied, or even failed to affirm,
some of the other axioms of ZF might still be called a set theory, albeit a
deviant or fragmentary one. But a theory that did not affirm that the
objects with which it dealt were identical if they had the same members
would only by charity be called a theory of sets alone [Boolos’s italics].
The important point here, regardless of analyticity, is that one feature we
look for in identifying something as a set theory is that its objects should
be identical when they have the same members. This seems correct to me.
Much of the value gained by sets is that they have a very clear identity cri-
terion in extensionality, and without this, it seems that we are dealing
with some other kind of entity. But non-well-founded set theories do
share this feature with the iterative conception. So if we pick out exten-
sionality as characterizing sets, then why should we favor the iterative con-
ception, especially as this key axiom does not follow from it? Indeed, we


Boolos, “The Iterative Conception of Set,” pp. –.

Ibid., p. .
 The Iterative Conception and Set Theory 
could, according to Boolos, deny any of the other axioms and still end up
with a fragmentary or deviant set theory.
This does not seem entirely correct. Throughout the history of set the-
ory, another axiom has also been cited as characterizing set theory: the
comprehension axiom. This also does not follow generally from the itera-
tive conception, but this is surely a virtue. In its most general formula-
tion, it should not be part of any set theory, since unrestricted
comprehension yields the set-theoretic paradoxes. However, the iterative
conception does yield separation, ZF’s restricted version of comprehen-
sion, and any set theory will need some such principle of set specifica-
tion. Without it, we can not specify any sets. This boils set theory down
to two identifying characteristics – extensionality and comprehension – and
nothing about these two principles suggests the iterative conception.
Indeed, they seem more likely to work against the iterative conception in
the following way.
The naïve conception of set gave us too much in that it yielded con-
tradictory sets, such as Russell’s set of all non-self-membered sets. One
fairly common way of judging a successful axiomatization is that it rules
out these contradictory sets while doing as little damage as possible to
the original theory. In this sense, the iterative conception as expressed by
ZF perhaps rules out too much. While we do not want contradictory


There is a potential difficulty for non-well-founded sets and the extensionality axiom. Set theory
with the regularity axiom assures us that there is a recursive algorithm for deciding identity
between sets. While non-well-founded sets do not violate extensionality, this axiom must be
somehow strengthened to account for identity between such sets. There are various ways in which
this might be done, but it is not entirely clear that any one of them is to be preferred. Aczel raises
this issue in his Non-Well-Founded Sets in considering the various anti-foundation axioms. For a
concise exposition of the issue, see W. D. Hart, “On Non-Well-Founded Sets,” Crítica XXIV:
(), pp. –. For a recent account of non-well-founded set identity in terms of games that is
fairly analogous to the situation with well-founded sets, see Forster, “The Iterative Conception of
Set,” pp. –. Quine himself recognized this issue at least as early as  in his “On the
Individuation of Attributes,” in Theories and Things (Cambridge, MA: Harvard University Press,
), pp. –. Here, he observes, “The system of my ‘New Foundations’ does have
ungrounded classes, and so the system of my Mathematical Logic; and it could be argued that for
such classes there is no satisfactory individuation. They are identical if their members are identical,
and these are identical if their members are identical, and there is no stopping. This, then, is a
point in favor of the systems that bar ungrounded classes” [Quine’s italics].

Rieger argues similarly in his “Paradox, ZF, and the Axiom of Foundation,” p. . In a sense, per-
haps something like this can be found already in Zermelo’s “Investigations in the Foundations of
Set Theory I” in his comment that in light of the paradoxes “there is at this point nothing left for
us to do but to proceed in the opposite direction and, starting from set theory as it is historically
given, to seek out the principles required for establishing the foundations of this mathematical dis-
cipline. In solving the problem we must, on the one hand, restrict these principles sufficiently to
exclude all contradictions and, on the other, take them sufficiently wide to retain all that is valu-
able in this theory” (p. ).
 Quine, New Foundations, and the Philosophy of Set Theory
sets, we also had other sets in the original theory, such as the universal set
specified by the instance of comprehension “(∃y)(∀x)(x ∈ y ↔ x = x).”
ZF does away with this, as does type theory, at least in its most general
version. (Recall that type theory does yield a series of quasi-universal sets.)
However, NF does not while still apparently ruling out contradictory sets.
Indeed, its only two axioms are versions of extensionality and comprehen-
sion. This is the same for type theory, but it has perhaps the undesirable
complication of dividing the universe into levels, again perhaps moving
beyond the most basic characteristics of sets. ZF’s axioms seem perhaps
even more ad hoc. Given the weakening of comprehension in the form of
separation, many of the other axioms just serve as fixes for restoring the
power that was lost from the original naïve theory. Starting with the itera-
tive conception and moving to ZF does seem to provide some motivation
for adopting ZF, but what I am trying to bring out here is that starting
from our naïve conception of set and moving back to ZF makes ZF seems
quite arbitrary and unmotivated as a set theory, at least with regard to cap-
turing the original notion of set with which we began. I do not take any
of these arguments to be definitive, but I do think that they show that
any set theory can perhaps be made to look quite arbitrary, depending on
what we take as our starting point for thinking about sets. We have seen
this play out in a variety of ways, particularly in investigating the historical
development of set theory.
This still leaves us to discuss the axiom of choice. I will postpone this
until the Chapter , as it will lead us to much more general considera-
tions of NF. For now, I will just sum up our considerations so far of
Boolos’s account of the iterative conception of set. The general point
that I have tried to make throughout this chapter is that while other set
theories might seem artificial, there is something quite artificial in ZF as
well. We have seen this both in the assumption of replacement as an
extra postulate and in ZF’s ruling out certain seemingly legitimate and
perhaps quite interesting sets, namely, the non-well-founded sets. Let us
turn now to the axiom of choice, which will lead us to consider more
generally NF and Quine’s views on set theory as a whole.
 CHAPTER 

New Foundations, the Axiom of Choice,

and Arithmetic

In this final chapter, we consider W. V. Quine’s New Foundations (NF)

more generally in comparison with other set theories, and we conclude
with a return to his views on the philosophy of set theory as a whole.
Recall from Chapter  that George Boolos claimed that ZF is the only
apparently consistent and independently motivated set theory. I have been
arguing against this claim, trying to point out the ways in which ZF is far
less intuitive than Boolos presents it and the ways in which other set the-
ories, including NF, are also independently motivated. Indeed, throughout
this book, I have been suggesting that the development of set theory has
been largely a battle of competing intuitions over what sets are like. This
may be worrisome in that it perhaps leaves us with no objective way of
determining what should count as a viable set theory. What we call set
theory could just be a matter of personal preference. Should we allow just
any theory of collections count as a set theory? However, the situation is
not as bleak as that. There do seem to be some readily available criteria
for assessing whether to count some body of theory as set theory.
We have already seen some of this in our recounting of the history of
the development of set theory in that the concept of a set began with
some idea of sets being the extensions of predicates (though initially,
Georg Cantor says little about what sets are). While I take all set theories
to have this origin – and the history seems to support this view – this in
itself is perhaps a controversial claim. Many philosophers and logicians
have stated that there are two separate notions of a set: the logical and
the mathematical. The former takes sets to be the extensions of predi-
cates, and the latter takes them to be formed according to the iterative
conception. Having touched upon this debate elsewhere, I will not say


Kurt Gödel, for example, draws this distinction in his “What Is Cantor’s Continuum Problem?” in
Philosophy of Mathematics, Selected Readings, eds. Paul Benacerraf and Hilary Putnam, nd edn
(New York, NY: Cambridge University Press, ), pp. –.


 Quine, New Foundations, and the Philosophy of Set Theory
more here other than to reiterate that I think that the distinction does
not hold up very well and that it is mostly a distinction that encourages
a privileging of the iterative conception. The usual argument is that the
logical conception of set has its home in philosophy and that is where
the paradoxes arose. Within mathematics, the iterative conception was
always present, so mathematicians were never troubled by the paradoxes
or the notion of a set more generally. However, the history shows the
iterative conception and this distinction emerging only after the para-
doxes (see Chapter  especially). I will now put this issue aside.
We saw in Chapter  that extensionality and comprehension give us
some minimal account of what to count as a set. As we saw in previous
chapters, I have suggested some additional criteria that any viable set the-
ory should meet. First, set theory must deal with the paradoxes in some
way. Second, set theory should contain a mathematically interesting
account of the infinite, as this was one of the aspects that drove the
development of set theory in the first place. Therefore it should preserve
at least some version of Cantor’s theorem. Third, set theory should be
capable of serving as a framework within which ordinary mathematics
can be carried out. This seems a reasonable, minimal set of criteria for
identifying a theory of collections as a viable set theory in the way the
tradition intended set theory to be, although we might still revise this list
in light of further development in set theory. We have already seen that
NF succeeds on the first two criteria. The primary focus of this
chapter will be to show that it succeeds also on the third. To begin this
discussion, I pick up with Boolos’s considerations of the axiom of choice
in relation to the iterative conception.

. The Axiom of Choice

Boolos introduces choice in the form often called “the multiplicative
axiom,” which states,
For any x, if x is a set of nonempty disjoint sets, then there is a set, that
is, a choice set for x, that contains exactly one member of each of the
members of x.


W. D. Hart suggested these to me in conversation, but Quine himself suggests something like this
in his “Foundations of Mathematics,” in The Ways of Paradox and Other Essays, rev. and enlarged
edn (Cambridge, MA: Harvard University Press, ), pp. –.

W. D. Hart has pointed out to me that the name comes from Bertrand Russell, as he used this ver-
sion of the axiom to show infinite products to be nonempty. For details, see Russell, Introduction
to Mathematical Philosophy (London: George Allen and Unwin, ), p. .
 New Foundations, the Axiom of Choice, and Arithmetic 
The axiom is crucial for much of ordinary mathematics, but as Boolos
rightly notes, neither choice nor its negation follows from the iterative con-
ception. Much like replacement, he suggests, we might extend the stage
theory so that it does decide choice (no doubt preferably that it be true).
“But,” he remarks,
it seems that no additional axiom, which would decide choice, can
be inferred from the rough description without the assumption of the
axiom of choice itself, or of some equally uncertain principle, in the
inference. The difficulty with the axiom of choice is that the decision
whether to regard the rough description as implying a principle about
sets and stages from which the axiom could be derived is as difficult a
decision, because essentially the same decision, as the decision whether
to accept the axiom.
For example, we might try to extend the stage theory by adding the fol-
lowing principle:
Let x be a set of nonempty disjoint sets, so x is formed at some stage s.
Then the members of x must be formed at some stage earlier than s.
Therefore at stage s or earlier, there exists a set that contains exactly one
member of each member of x.
But such a principle, Boolos explains, begs the question, since we have
no reason to think that such a set is in fact formed. It is the axiom of
choice itself that allows us to choose exactly one member from each
member of x, that is, to form a choice set for x. He concludes, “To say
this is not to say that the axiom of choice is not both obvious and indis-
pensable. It is only to say that the justification for its acceptance is not to
be found in the iterative conception of set.”
Again, as with the axiom of replacement, a crucial mathematical prin-
ciple does not follow from the iterative conception of set. This is again
not to say that we should reject ZF as a framework for mathematics;
rather, we can get only so far motivating it by way of the iterative con-
ception. The situation is the same for type theory; choice must be
assumed as an additional axiom if the system is to be sufficient for ordin-
ary mathematics. If we approach set theory in a pragmatic spirit, such
additional postulates should be of minimal concern. Our guide to devel-
oping set theory is to incorporate enough power to make the system
mathematically interesting. But the option of adding choice is not

Boolos, “The Iterative Conception of Set,” p. .

Ibid., pp. –.

Ibid., p. .
 Quine, New Foundations, and the Philosophy of Set Theory
available to NF. In fact, this will be our first serious challenge in adopt-
ing NF as a set theory, for NF does decide choice, and it says that this
principle is false. So can NF serve as a plausible framework for mathe-
matics? This is the question to which we now turn.
There is no doubt that the axiom of choice is an important principle
for mathematics, used in a wide variety of results from all areas of the
discipline. But the axiom itself has had a rather controversial history.
Ernst Zermelo, in his  well-ordering paper, was the first to put for-
ward the principle as an explicit axiom, which he used to show that every
set can be well ordered. Attacks on the proof immediately followed,
describing its use as an illegitimate method for mathematics. Zermelo
responded with his  proof and his fully developed axiom system for
set theory. Although choice finally won out as an accepted method of
proof, its status remains somewhat different from that of other axioms of
set theory. Justification for the axiom generally points to its usefulness
and importance in mathematics. Indeed, much of Zermelo’s strategy in
defending it was to show that a significant amount of already accepted
mathematics relies upon it, albeit as an implicit assumption. Such argu-
ments continue to be common in set theory texts. For example, Jech
points to important results that rely on choice such as the Hahn-Banach
theorem, Tichonov’s theorem, and the maximal ideal theorem as
accounting for the axiom’s “universal acceptance.” Despite this wide-
spread acceptance, mathematicians and logicians continue to be careful
about the use of choice. In particular, they often distinguish proofs in
ZF from those done in ZFC (ZF plus choice) and frequently try to re-
prove results that rely on choice with proofs that do not. Indeed, much
of Jech’s most notable work focuses on such results. Quine himself,
perhaps unsurprisingly, adopts this attitude in Set Theory and Its Logic,
remarking,
Intuitively the principle seems reasonable, even obvious. The trouble is
just that it is not as simple and elementary a statement as could be desired
as a starting point, and no way is known of deducing it strictly from


For some examples of the mathematics relying on choice, see Hrbacek and Jech, Introduction to
Set Theory, pp. –.

Ernst Zermelo, “Proof That Every Set Can Be Well-Ordered,” pp. –.

For a complete history of the axiom and the controversies that followed it, see Gregory H. Moore,
Zermelo’s Axiom of Choice: Its Origins, Development, and Influence (New York, NY: Springer, ).

Indeed, Moore argues that the controversy over choice was the primary motivation for Zermelo’s
axiomatization. As I stated in Chapter , I do not fully agree with this account.

Hrbacek and Jech, Introduction to Set Theory, p. .

Thomas J. Jech, The Axiom of Choice (Mineola, NY: Dover, ).
 New Foundations, the Axiom of Choice, and Arithmetic 
anything simpler. For this reason it is common practice to distinguish
between results that depend on the axiom of choice and those that do
not. Accordingly I shall not adopt the principle as axiom or axiom
schema, but shall merely use it as premiss where needed.
Aside from such justifications by way of its usefulness, there may also be
something that seems rather obvious about the axiom. From a collection
of disjoint sets, it allows us to pick a unique representative for each one,
yielding a choice set for the collection. Jech observes that the axiom is
often described by using the analogy of an election in which the sets in
the collection are the candidates for a particular office and the election
process gives us a choice function from each of these sets. But he is careful
to point out that such an analogy talks only of a finite case and that this
gives us no justification for assuming the axiom to hold in infinite cases.
Reflecting on the arithmetic of infinite numbers, for example, suggests in
a pretty straightforward way that what holds for the finite should give us
no reason to think that it will hold analogously for the infinite. Indeed, as
I have pointed out several times, we seem to have very few initial correct
intuitions about the infinite, and it was only Cantor’s discovery of set the-
ory that allowed us to gain an understanding of it.
It turns out that for the finite case, the axiom is unnecessary, since it
is provable in dealing only with finite collections of finite sets. The
situation is, of course, very different for infinite sets. Indeed, even when
an infinite collection contains only finite sets, choice is not generally
provable. So choice must be added as an axiom to deal with infinite
sets in general. Now we get some fairly unintuitive results. Stefan
Banach and Alfred Tarski, generalizing a result of Felix Hausdorff,
proved, using choice, that a sphere can be decomposed into parts and
rearranged, using rigid motions, to obtain two spheres, each of which is
the same size as the original sphere. A popular gloss on the result is
that it allows us to decompose a sphere the size of a pea and reassemble


W. V. Quine, Set Theory and Its Logic. Rev. edn (Cambridge, MA: Harvard University Press,
), pp. –.

Thomas J. Jech, “About the Axiom of Choice,” in Handbook of Mathematical Logic, ed. Jon
Barwise (Amsterdam: North-Holland, ), p. .

Ibid., pp. –.

Ibid., pp. –. The original paper is Stefan Banach and Alfred Tarski, “Sur la Décomposition
des Ensembles de Points en Parties Respectivement Congruents [],” in Alfred Tarski,
Collected Papers, vol. I, –, eds. Steven R. Givant and Ralph N. McKenzie (Boston, MA:
Birkhäuser, ), pp. –. Many authors mention this result to illustrate that choice is not
as intuitive as it may seem. There are other results of this kind, such as the existence of unmeasur-
able sets of real numbers, but such examples are not nearly as vivid for the nonspecialist.
 Quine, New Foundations, and the Philosophy of Set Theory
it into a sphere the size of the sun. This has become known as the
Banach-Tarski paradox. The point here is not to claim that choice must
be false in light of such a result, but merely to bring out its lack of
grounding in common sense. As Michael Potter observes, the unintui-
tive nature of the result depends upon our trying to understand it in
terms of our ordinary intuitions about elementary geometry. The argu-
ment for this paradoxical result does not rely on any such intuitions but
rather depends on a mathematical understanding of transcendental
functions.
I have been continually coming back to such attempts at grounding
set theory in ordinary everyday conceptions of, say, collections or of the
infinite. Our understanding of such issues may develop in conjunction
with certain initial intuitions about sets, but this involves a continual
back and forth between the technical development of the subject and
our initial intuitions about it. If intuitions alone were the final arbiter for
the success of a mathematical theory, we would have to throw out much
of mathematics as we now know it. Certainly set theory would have
ended with Bertrand Russell’s paradox. I also want to make the point
that there is some unfairness in examining set theories other than ZF (or
ZFC). Because they violate our ordinary understanding of, say, what it
means to be a member of a collection, they are often treated with disdain
and as being unworthy as research projects. I am suggesting that such
nonstandard set theories are thought to be nonstandard, that is, contrary
to intuition, only because we have become so familiar with the iterative
conception as exemplified by ZF.
To return to our more immediate topic, what of the axiom of choice
in NF? The situation does not initially look good, since Ernst Specker
proved in  that choice is refutable in NF. We want to know
exactly what Specker proved, though. Will it prohibit much of accepted
mathematics? If so, this seems to be a serious point against NF as a fra-
mework within which mathematics can take place. If much of accepted
mathematics can be developed in NF, however, this cannot be used as a
point against NF. In fact, we might then have reason to prefer NF over


For a modern technical account, see Stan Wagon, The Banach-Tarski Paradox (Cambridge:
Cambridge University Press, ). This gloss can be found on p. xiii.

Michael Potter, Set Theory and Its Philosophy (New York, NY: Oxford University Press, ),
pp. –.

Ernst P. Specker, “The Axiom of Choice in Quine’s New Foundations for Mathematical Logic,”
Proceedings of the National Academy of Sciences, : (September ), pp. –.
 New Foundations, the Axiom of Choice, and Arithmetic 
ZF. Choice is independent of the axioms of ZF; we can neither prove it
nor refute it in ZF. On this count, then, NF seems to be a more ideal
theory, as Boolos characterizes an ideal theory, since it decides the axiom
of choice, whereas ZF does not. If we were to accept NF as a working
set theory, we would get both the development of accepted mathematics
and an answer to the truth or falsity of choice. Let us see, then, what
exactly the situation is.
Specker’s refutation of choice in its original form is rather abstract and
hard to grasp apart from what appear to be some rather arcane details of
NF. To make the proof more accessible, I instead follow the account
that J. Barkley Rosser gives in his review of Specker’s result. We will
assume the axiom of choice and then proceed by reduction. Rosser
begins by observing that the proof depends significantly on many proper-
ties of cardinals being invariant when their types are raised uniformly.
That is, given a cardinal m, we choose a class α of cardinality m and let
T(m) be the cardinality of the singleton subclasses of α, USC(α). T(m)
thus gives us the cardinal m, but it is of one type higher. (We saw the
importance of such type raising operations in NF in Chapter  in the
discussion of Cantor’s theorem.) To illustrate such invariance, Rosser
gives a few examples:
m ≤ n ≡ T(m) ≤ T(n);
T (m + n) = T(m) + T(n), when m + n, m, and n are all cardinals; and
T() = , T() = , T() = , and so on.
As a consequence, we see that if m = q + q + q + k, where k = , , or ,
then T(m) = T(q) + T(q) + T(q) + k. So while residues (mod ) are
invariant, cardinality is in general not. In fact, we cannot even prove for
all finite m that T(m) = m.
Next, to define m so that it has the same type as m, we find an α, if
such an α exists, such that the cardinality of USC(α) = m. We then let m
be the cardinality of all subclasses of α. Now if m exists, then m < m and
T(m) = T(m), another case of invariance. However, m does not exist for
all m. Indeed, if |V| is the cardinality of the universe, then m exists if and
only if m ≤ T(|V|). We also note that T(|V|) = |V|, T(T(|V|)) = T(|V|), and
so on. We also define, for many cardinal m, Φ(m) as the set (it is stratified)
of m and as many of m,  … as exist. If they all mexist so that Φ(m) is infi-
nite, then Φ(T(m)) consists of T(m), T(m), T( ), … and is also infinite.


J. Barkley Rosser, “Review of Specker’s ‘The Axiom of Choice in Quine’s New Foundations for
Mathematical Logic,’” Journal of Symbolic Logic, : (), pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
The interesting case is when Φ(m) is finite. For example, |V| does not exist,
so Φ(|V|) contains only |V|.
By choice, we can show that if Φ(m) is finite, then Φ(T(m)) is also
finite and also that the cardinalities of Φ(m) and Φ(T(m)) have different
residues (mod ). Let Φ(m) have finite cardinality. Then there is a great-
est member of Φ(m), say, n. So n cannot exist; otherwise, it would be
the greatest member of Φ(m). Thus T(|V|) < n, since cardinal com-
parability follows from choice. Then T(T(|V|)) < T(n), so T(|V|) =
T(T(|V|)) ≤ T(n). This gives us two cases to consider:
Case : T(|V|) = T(n). ThenT(n)the members of Φ(T(m)) are T(m),
T(m), …, T(n), T(n),  , so intuitively, Φ(T(m)) has two more
members than Φ(m). So Φ(T(m)) is finite, and its residue (mod )
differs from that of Φ(m).
Case : T(|V|) < T(n). Then T(n) is the greatest member of Φ(T(m)).
So now Φ(T(m)) has intuitively one more member than Φ(m), and,
similar to the first case, Φ(T(m)) is finite, and its residue (mod )
differs from that of Φ(m).
By choice again, there is a least m for which Φ(m) is finite, and since
Φ(T(m)) is also finite, m ≤ T(m). By the definition of T(m), there is a p
such that m = T(p) and p ≤ m. Hence p = m, so T(p) = T(m), and then
finally m = T(m). Therefore the cardinals of Φ(m) and Φ(T(m)) have the
same residue (mod ), which is a contradiction. Thus the axiom of
choice is false in NF. Also, since the generalized continuum hypothesis
implies choice, it too must be false.
Where does this leave us with regard to NF as a framework for ordin-
ary mathematics? It seems to leave us in a situation that is perhaps sur-
prisingly similar to where we are with ZF. Rosser observes almost
immediately in his note that he “has become increasingly convinced
that” all ordinary mathematics can be developed within the Cantorian
sets, that is, those sets α such that they and the set of their singleton sub-
sets have the same cardinality. (We discussed such sets in Chapter 
with reference to Cantor’s theorem.) For such sets, it seems quite reason-
able to assume that choice holds. Indeed, Specker’s proof shows it failing


Adolf Lindenbaum and Alfred Tarski, “Communication sur les Recherches de la Théorie des
Ensembles [],” in Alfred Tarski, Collected Papers, vol. I, eds. Steven R. Givant and Ralph N.
McKenzie (Boston, MA: Birkhäuser, ), pp. –. They sketch their result on pp. –.
Waclaw Sierpinski proved it in “L’hypothèse généralisée du continu et l’axiom du choix,”
Fundamenta Mathematicae  (), pp. –.

Rosser, “Review of Specker,” p. .
 New Foundations, the Axiom of Choice, and Arithmetic 
only for non-Cantorian sets such as the universal set. Quine says that
we might think of the situation as follows:
So what we have is just another case, and a beauty, of unconventional
behavior on the part of non-Cantorian classes … Classical results can still
be got in NF by inserting, where needed, a premiss that a class concerned
is Cantorian. One could look upon NF as merely more general, in this
respect, than set theories where everything is Cantorian.
Furthermore, the possibility that these non-Cantorian sets might yield
important results for the Cantorian sets should not be overlooked:
The demonstrability of “Λ ∉ N” [a version of the axiom of infinity] sug-
gests that non-Cantorian classes, with all their unconventional behavior,
are not just a harmless annex to an otherwise well-behaved universe, but a
valuable intermediary in proving desirable theorems about the well-
behaved objects themselves. For it is the failure of the axiom of choice for
non-Cantorian classes that has enabled Specker to prove “Λ ∉ N”, which
itself says nothing of non-Cantorian classes.
Quine’s remark here is particularly interesting because it seems rather
close to the kind of reasoning that is often provided by set theorists
working in the ZF tradition trying to justify large cardinal hypotheses.
While such cardinal numbers might initially appear too large to have
anything to do with ordinary mathematics – never mind natural science –
it turns out that they often do have implications for the more ordinary,
undisputed, lower-level mathematics. Quine is here suggesting an analo-
gous situation with regard to the big sets of NF such as the set of cardi-
nal numbers, the set of ordinal numbers, and the universal set itself.
As I have already remarked, mathematicians are generally careful about
their use of choice, not necessarily doubting its truth but still keeping
track of where it is necessary to a result. Rosser is no exception. Even


We should also note that NF is not the only set theory for which AC fails. The axiom of determi-
nacy has often received serious consideration as an additional axiom to add to ZF-style set the-
ories, but as with NF, such theories also disprove AC. For more on this topic, see Akihiro
Kanamori, The Higher Infinite, nd edn (New York, NY: Springer, ), pp. –.

Quine, Set Theory and Its Logic, p. .

Ibid., p. .

Although he does not cite this comment from Quine himself, Thomas Forster also makes this
point in his Stanford Encyclopedia of Philosophy entry “Quine’s New Foundations”: https://plato
.stanford.edu/entries/quine-nf/. For some indication of the way in which large cardinal hypotheses
have consequences for lower level mathematics, see Hrbacek and Jech, Introduction to Set Theory,
pp. –.

In a way, Quine’s work in Set Theory and Its Logic can be viewed as a generalization of this
approach to set theory.
 Quine, New Foundations, and the Philosophy of Set Theory
before Specker’s proof, when it might have seemed completely reasonable
to assume the full axiom of choice, that is, choice for the universe,
Rosser had already limited himself in his  Logic for Mathematicians,
which takes NF as its basis, to the denumerable axiom of choice: the
existence of a choice function for countable families of nonempty sets.
Indeed, denumerable choice suffices for the development of most mathe-
matics, including that of the real numbers. As Rosser describes the
situation in a post-Specker appendix to Logic for Mathematicians, there
are an infinite number of choice axioms, each of which says that we can
make λ many choices, where λ is any cardinal number. It is only where
λ is the cardinality of the universe that Specker’s proof holds. Hence
this result does not threaten Rosser’s reduction of mathematics to NF.
The failure of choice for NF generally may still seem to be an oddity,
perhaps an uncomfortable one, since appeal to it has become so
ingrained in contemporary mathematical practice. While Zermelo’s well-
ordering theorem claimed to show that all sets can be well ordered, the
theorem cannot hold in NF, since this is equivalent to the axiom of
choice. But we should not forget that Zermelo’s set theory rules out cer-
tain sets from the start, most noticeably – in comparison to NF – the
universal set. And the universe is such a set that is non-well-orderable.
If our starting point does not allow for such sets, then there are no pro-
blems for Zermelo’s well-ordering principle. But the very question that I
have been trying to raise throughout this book is about what grounds
there are for ruling out sets such as the universal set from the start. It
seems to have a perfectly clear specifying condition, and it also seems
perfectly reasonable that in the course of set-theoretic research (because
our interest is in sets as such), we might want to know what the universe
and other big sets are like. Indeed, as we have seen, this was one of
Quine’s driving motivations in proposing NF in the first place. With the
proof of the axiom of infinity, we saw an example of how these big sets
might yield insights into the more conventional sets that are present in
the more common axiomatizations of set theory. Why not just do as
Quine urged, then, and think of NF as merely a more general theory,
one that includes sets that ZF leaves out?

J. Barkley Rosser, Logic for Mathematicians, nd edn (Mineola, NY: Dover Publications, ),
p. .

For some discussion and examples, see Potter, Set Theory and Its Philosophy, pp. –.

Rosser, Logic for Mathematicians, pp. –.

However, the universe can be well ordered in NFU as long as there are enough atoms (urele-
ments) in the universe. For details, see Randall Holmes “The Set-Theoretical Program of Quine
Succeeded, but Nobody Noticed,” Modern Logic  (), pp. –.
 New Foundations, the Axiom of Choice, and Arithmetic 
It should be noted that NF is not the only set theory to disprove the
axiom of choice. This occurs also in ZF-style set theories to which the
axiom of determinacy has been added. To gain some understanding of
the axiom, start by thinking of a real number as being represented by an
infinite sequence of s and s. A pair of players in an infinite game
then take turns playing a  or a , which will generate a real number.
Now, if A is a set of real numbers, there is a game G(A) associated with
this set. The first player will win the game if the real number generated
by the two players is in A, and the second player will win if it is not.
This game and the set A are said to be determined if and only if one of
the players has a winning strategy for the game. The axiom of determi-
nacy (AD) states that every set of real numbers is determined. This
axiom might seem to lack much in the way of intuitive motivation as a
natural candidate for being added to the axioms of ZF. But by a theorem
of Jan Mycielski and Stanislaw Swierczkowski, it does imply some very
desirable properties for sets of real numbers, namely, that every set of
real numbers is Lebesgue measurable, has the Baire property, and has the
perfect set property. However, AD also contradicts the axiom of
choice, since choice implies the existence of an undetermined set of real
numbers. This was, however, much the point; AD was supposed to elim-
inate the undesirable nonmeasurable sets that result from full choice.
So here we have an additional axiom candidate for ZF that, like the
system of NF as a whole, denies the axiom of choice. In recent times,
largely owing to its denial of choice, AD has received less attention as a
plausible additional axiom for ZF. However, in an early monograph on
AD, Eugene Kleinberg argues in much the same way that I argued with
regard to the failure of choice in NF:
Contradicting the axiom of choice is not, in and of itself, grounds for
rejecting the axiom of determinateness. Set theorists long ago segregated
the axiom of choice from the standard axioms of set theory due to its

In my exposition here, I follow that found in Penelope Maddy, Naturalism in Mathematics
(Oxford: Clarendon Press, ), p. . Hrbacek and Jech’s account in their Introduction to Set
Theory, pp. –, is more detailed and slightly more technical but is also quite accessible.

There are variations in how the axiom is stated. Kanamori also states the axiom this form in his
The Higher Infinite, p. . Chapter  of his book provides a thorough technical treatment of
determinacy as well as its historical setting. For further details, see also Yiannis Nicholas
Moschovakis, Descriptive Set Theory (New York, NY: North-Holland, ), chapter .

This is stated in Kanamori, Higher Infinite, p. , along with some useful historical context. The
original paper is Jan Mycielski and Stanislaw Swierczkowski, “On the Lebesgue Measureability
and the Axiom of Determinateness,” Fundamenta Mathematicae  (), pp. –. It should
be noted that in earlier discussions of AD, it tended to be called the axiom of determinateness.

Kanamori, The Higher Infinite, pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
somewhat tenuous nature. Mathematicians in general had been aware of
problems with the axiom of choice, and following such things as the
Banach-Tarski paradox, analysts avoided it whenever possible.
Most research on determinacy has now shifted to the axiom of projective
determinacy, which still yields the desirable properties for sets of real
numbers but does not deny the axiom of choice. In a further connection
with NF, AD does still allow for weaker forms of choice. Rather than
denying AD, Jech, in his standard work on set theory, adopts this strat-
egy of assuming a weakened form of choice. Because AD has large car-
dinal consequences, one might start to wonder whether there is just no
reason to expect choice to hold as we get closer and closer to obtaining
the entire universe of set theory.
As a final point on the axiom of choice, recall Boolos’s comment about
an ideal theory: that it should decide the continuum hypothesis, which is
independent of the axioms of ZF (and ZFC). Choice is also independent
of ZF and has had a controversial history with regard to its truth. Most
logicians and mathematicians now accept it but largely, as we have seen,
because of its pragmatic value for mathematics overall; it yields many
desirable results and no (or perhaps not many) undesirable ones. But NF,
too, can have choice – just not in the unrestricted form in which ZFC has
it. Again, this is because ZFC is already restricted in its banning of such
sets as the set of cardinal numbers, the set of ordinal numbers, and the
universal set itself. Restricted versions of choice, such as the denumerable
axiom of choice or choice restricted to Cantorian sets, will work perfectly
well in NF and provide for much of ordinary mathematics, just as ZFC
does. Yet NF is perhaps more ideal (in Boolos’s sense) than ZF with
regard to choice. Choice is independent of ZF and so must be added as
an additional axiom if ZF is to serve as a framework for mathematics. In a
sense, then, ZF gives us no reason to think that choice is true. Choice
yields many desirable consequences, so it makes sense that we would want
it to be true. But wanting something and its actually being so are two dif-
ferent things. NF tells us that choice in its most general form is false and
so actually decides a set-theoretic statement that ZF does not. This does


Eugene M. Kleinberg, Infinitary Combinatorics and the Axiom of Determinateness. Lecture Notes in
Mathematics  (New York, NY: Springer-Verlag), p. .

Thomas Jech, Set Theory, rd edn (New York, NY: Springer-Verlag, ), p. .

Potter, Set Theory and Its Philosophy, pp. –, includes a discussion of the point at which, in
our assumption of larger and larger large cardinal axioms, we have to give up choice.

Again, see Moore’s Zermelo’s Axiom of Choice for a complete history of the axiom, including the
controversies that it raised.
 New Foundations, the Axiom of Choice, and Arithmetic 
not inhibit the development of mathematics in NF, since choice holds for
Cantorian sets. Indeed, we might think of this failure as telling us some-
thing further about all the sets there are. In ZF-style set theory, we build
in the assumption that all sets are Cantorian. With NF, we explore the
set-theoretic universe as a whole without prejudice toward particular kinds
of sets. In the failure of choice, we then gain some insight into the general
properties of non-Cantorian sets and what distinguishes them from
Cantorian sets. Furthermore, as a consequence of disproving choice, we
also gain information about the generalized continuum hypothesis
(GHC). A result of Tarski and Adolf Lindenbaum shows that GHC
proves choice, so GHC is also false in NF. So in Boolos’s sense, it seems
that NF may be a more ideal theory than ZF (or ZFC) in its ability to
decide questions such as these.

. Arithmetic in New Foundations

Let me return now to more mundane matters of arithmetic. That the fail-
ure of choice in NF does not inhibit the development of ordinary mathe-
matics in this theory is reassuring, but this might not be the only worry to
have. As is well known, developing mathematics set-theoretically generally
proceeds by first showing that set theory captures arithmetic. The other
number systems of integers, rational numbers, and real numbers are then
built up from this basis. So we also want to be certain that NF can
provide for arithmetic, and here, further difficulties arise. Because NF’s
comprehension schema is restricted by stratification, certain inductive defi-
nitions that are used in arithmetic will be problematic. In this sense, then,
despite allowing for big sets, NF is more restrictive than ZF.


Forster, in his “The Status of the Axiom of Choice in Set Theory with a Universal Set,” Journal of
Symbolic Logic : (), pp. –, tries to sketch out some general characteristics of where and
why AC fails for set theories with a universal set. There are, of course, theories related to ZF for
which AC fails, e.g., ZF + Determinacy. It seems to me that this may be one particularly good rea-
son for taking research on NF seriously: Perhaps we will find trends across set theories that will
help us to see connections rather than differences – in a sense, to come to a better understanding of
a unified body of knowledge called set theory rather than ZF-style set theory (which is usually taken
to be set theory simplicter) and its deviant relatives such as NF. This seems to be perfectly in line
with a Quinean approach to set theory as a comparative endeavor investigating whatever sets there
might be rather than just our favorite ones. In Set Theory and Its Logic, Quine himself describes the
failure of choice not as a mark against NF but rather as “just another case, and a beauty, of uncon-
ventional behavior on the part of non-Cantorian classes … One could look upon NF as merely
more general, in this respect, than set theories where everything in Cantorian” (p. ).

Lindenbaum and Tarski, “Communication sur les Recherches de la Théorie des Ensembles, and
Sierpinski, “L'hypothèse généralisée du continu et l'axiom du choix.”

NF is in much the same situation as type theory on this point.
 Quine, New Foundations, and the Philosophy of Set Theory
For NF, sets cannot be specified when their specifying conditions are
unstratified, and this comes into play with induction, since we will then
have induction only for stratified conditions. So, for example, we might
expect that we should be able to prove for each natural number n that it
counts the set of its predecessors. That is, we might expect to prove by
induction (∀n)(n ∈ N ⊃ {m: m ∈ N .  < m ≤ n} ∈ n, but we see that
this condition is not stratified, since n occurs on both sides of the mem-
bership relation. Therefore we will not be able to state the existence of
such a set in NF.
To resolve this sort of issue, Rosser put forward the axiom of counting,
which states that for any natural number n, n counts the number of its
predecessors, (∀n ∈ N){m:  < m ≤ n} ∈ n. Equivalent to the axiom is
the type-raising operation T for ordinals (analogous to T for cardinals)
when it is assumed that T is the identity map when applied to natural
numbers, that is, for all natural numbers n, T{n} = n, or the order-type of
the relational unit subsets of n is equal to n. This further yields that all
finite sets are Cantorian, that all finite sets are strongly Cantorian (that is,
the singleton function is a set), and that there is an infinite strongly
Cantorian set. Indeed, these claims are equivalent under the axiom of
counting. The axiom of counting also helps to preserve a variety of
more ordinary mathematical results, such as the theorems of number the-
ory that concern the number of integers having a particular property. It
also simplifies the natural numbers generally in assuring us that if n is a
natural number, then for any set α ∈ n, α has n members.
But let us not forget that counting is an additional axiom that is
not deducible from NF’s two original axioms: extensionality and compre-
hension. We might try to justify it on the basis of its rather intuitive-
sounding nature, though we should be wary of such justifications in past

The natural numbers in NF are of the Frege-Russell variety, that is,  is the set of all sets with no
members,  is the set of all singletons,  is the set of all pairs, and so on.

For details, see Forster, Set Theory with a Universal Set: Exploring an Untyped Universe (Oxford:
Clarendon Press, ), pp. –; and Rosser, Logic for Mathematicians, pp. –.

For these and some other examples of counting in ordinary mathematics, see Rosser, Logic for
Mathematicians, pp. –.

Quine’s system Mathematical Logic – NF with proper classes added – does not face the problem
with induction but requires an additional axiom to guarantee that the natural numbers form a set.
Quine thought such an addition inelegant; see his “The Inception of ‘New Foundations’,” in
Selected Logic Papers, enlarged edn (Cambridge, MA: Harvard University Press, ), p. . No
doubt he would think the same about adding the axiom of counting to NF. He states this for sup-
plementing NF in other ways to secure induction; see Set Theory and Its Logic, p. . This sort
of dissatisfaction seems just to bring us back to Zermelo’s observation that no solution to the
paradoxes has been found that is “entirely satisfactory.” We will always be giving up something of
our most natural understanding of sets.
 New Foundations, the Axiom of Choice, and Arithmetic 
work, for example, with regard to choice. Indeed, much like choice,
counting is a surprisingly powerful axiom. In fact, NF plus counting
(NFC) proves the consistency of NF, so on pain of inconsistency (by
way of Gödel’s second incompleteness theorem), it had better not be the
case that counting follows from the two initial axioms. Still, we might
say that counting fares somewhat better than choice in not yielding any
especially counterintuitive results – aside from those that are already pre-
sent in NF without counting (and these results tend to come from the
presence of non-Cantorian or big sets in NF and so are perhaps counter-
intuitive only because other set theories do not allow us to investigate
such sets). Unless we demand some further self-evident grounding for
our mathematics, we seem to be no worse off in our choice of axioms
than we are with ZF (or ZFC). Indeed, in its economy of axioms, NF
recommends itself considerably over ZF (and ZFC). This is enough for
the point I have been arguing: that, so far at least, the view that ZF
somehow gives us the only viable account of sethood relies more on pre-
judice than on fact.
Despite being surmountable, the difficulty over induction that emerges
out of NF’s restriction to stratified conditions suggests that perhaps NF
should not be preferred as a foundation for ordinary mathematics. This
is no objection to my view of NF. My aim here has only been to show
that, like other set theories, NF can capture ordinary mathematics and
that, also like other set theories, its axioms will need supplementing to
do so. Even if we should still think that NF develops arithmetic in a
way that is much more unnatural than the way in which it is developed
in, say, ZF, this is just a tradeoff between the two set theories. What NF
gives us in return that ZF does not is a way into the mathematics of the
absolute infinite. This is a point to which I will return shortly.

. Exploring the Set-Theoretic Universe

In this concluding section, I want to try to provide some summary of what
exactly a Quinean philosophy of set theory looks like. Given what we
have seen of its exploratory and experimental nature, its purpose cannot be


Again, other set theories require an axiom of infinity, whereas NF requires the axiom of counting.

Joseph Ullian, in his overview of Quine’s work in logic and its relevance to his philosophy, has
influenced my understanding of Quine’s logical work greatly, not only in what follows immedi-
ately but throughout this book. See his “Quine and the Field of Mathematical Logic,” in The
Philosophy of W. V. Quine, eds. Edwin Hahn and Paul Schilpp, expanded edn (Chicago, IL: Open
Court Publishing, ), pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
to bring forth something like the essence of sethood as a sort of a priori
science. Quine is a pluralist about set theory, as I take it he is in some sense
a pluralist about all our scientific theories. This is not to say that almost
anything might count as set theory for him. Indeed, the point of
Section . was to show, according to some general and rather uncontro-
versial criteria, that Quine’s NF could be reasonably included within the
discipline of set theory. So, given his approach to set theory, I want to
return to the topic of what a Quinean philosophy of set theory is, if not to
bring forth the essence of sethood, for example, in terms of an iterative
hierarchy.
I argued in Chapter  that the culmination of Quine’s work in set the-
ory came in his  Set Theory and Its Logic, a work that, I claim,
stands alongside Word and Object as one of his greatest philosophical
achievements. Set Theory and Its Logic is dedicated to Russell for initiat-
ing Quine’s interest in the subject, and it has as its epigraph a quote
from Gilbert and Sullivan’s The Pirates of Penzance: “How quaint the
ways of paradox,” stressing again the central importance Quine sees in
the paradoxes as giving rise to the development of set theory. His major
theme is a comparative study of the main approaches to set theory. As
we saw in the first two parts of the book, Quine develops a set theory
that he intends as largely neutral between the usual systems of Russell,
Zermelo, and himself. He describes his policy as one of minimizing
assumptions about set existence, making them only where the develop-
ment of the theory demands them. To this end, he employs the appa-
ratus of virtual classes, simulating talk of classes by way of contextual
definitions rather than assuming their existence from the start. This gives
him a useful contrast later, when he introduces actual classes, better
demonstrating “what power real classes confer that the counterfeits
do not.” He adopts axioms that are strong enough only to imply the
existence of finite sets. Where more substantial claims enter in, such as
the existence of infinite sets or the axiom of choice, he adopts these only
as hypothetical claims at the outset of proving the theorem that requires
these added assumptions. It is only after he has introduced his neutral
and largely minimal set theory that he thinks we are ready, in the third
part of the book, to consider the relative advantages and disadvantages of


Quine, Set Theory and Its Logic; W. V. Quine, Word and Object (Cambridge, MA: MIT Press,
).

Quine, Set Theory and Its Logic, p. vii.

Ibid., p. ix.

For example, see Quine’s discussion of this approach in Set Theory and Its Logic, p. xi.
 New Foundations, the Axiom of Choice, and Arithmetic 
the various more familiar and substantial set theories. His aim is to pre-
pare us to consider the relative merits of the usual set theories from a
perspective that is prejudiced toward none of them.
Quine also returns to the theme that “intuition is bankrupt” when it
comes to developing a viable system of sets, for intuition brought set the-
ory in its earliest days to contradiction. This is not to say that intuitions
never come into play or are never helpful in set theory, but we need to
be cautious, as Quine explains (I put forth a similar argument in
Chapter , but here we have Quine himself making it):
The notion of class is so fundamental to thought that we cannot hope to
define it in more fundamental terms. We can say that a class is any aggre-
gate, any collection, any combination of objects of any sort; if this helps,
well and good. But even this will be less help than hindrance unless we
keep clearly in mind that the aggregating or collecting or combining here
is to connote no actual displacement of the objects, and further that the
aggregation or collection or combination of say seven pairs of shoes is not
to be identified with the aggregation or collection or combination of those
fourteen shoes, nor with that of the twenty-eight soles and uppers. In
short, a class may be thought of as an aggregate or collection or combina-
tion of objects just so long as “aggregate” or “collection” or “combination”
is understood strictly in the sense of “class.”
This is not to say that we have no way of knowing what a class is. Quine
does think that there is some use for coming to understand classes by way
of an analogy to the ordinary collections of physical objects we come in
contact with every day. But to understand collections as classes, he turns to
predicates. If we begin with a sentence about a thing, we can then think
of removing any reference to that thing from the sentence, leaving us with
only the predicate, or an open sentence, true of some things and false of
others. The class, then, as we now know, is exactly those things for which
the predicate is true – the extension of the predicate. Quine also notes
that we want classes to be identical when they have the same members, so
we also adopt extensionality as our principle of class identity. But he con-
tinues to be careful in appealing to intuitions; his gesturing at what a class
is as distinct from ordinary collections is not meant to be definitive:
I was describing the function of the notion of class, not defining class.
The description is incomplete in that a class is not meant to require, for

As Quine remarks, on p. x of Set Theory and Its Logic, “Because the axiomatic systems of set the-
ory in the literature are largely incompatible with one another and no one of them clearly deserves
to be singled out as standard, it seems prudent to teach a panorama of alternatives.”

Quine, Set Theory and Its Logic, p. .
 Quine, New Foundations, and the Philosophy of Set Theory
its existence, that there be an open sentence to determine it. Of course, if
we can specify the class at all, we can write an open sentence that deter-
mines it … But the catch is that there is in the notion of class no pre-
sumption that each class is specifiable. In fact there is an implicit
presumption to the contrary, if we accept the classical body of theory that
comes down from Cantor.
Maintaining Cantor’s theory of the infinite is not the only reason Quine
cites for not adhering too closely to the idea of classes as the extensions
of predicates. Because this “natural attitude,” as he describes it, led to
contradiction, we must instead make deliberate and careful choices about
our class existence axioms, since “intuition is not in general to be trusted
here.” But because we have a variety of interesting alternative axiomatiza-
tions available, Quine thinks that it hasty to focus on just one of them
“to the point of retraining our intuition to it.” In the contemporary
philosophy of set theory, this is exactly what has happened. Being a
viable set theory has become nearly inseparable from the intuitions guid-
ing the iterative conception of set.
In Quine’s view, then, seeming artificiality is not a reason to reject a
particular set theory. Yet he does not rule out the possibility that some
particular set theory will eventually be adopted as best, for he thinks that
his own approach – a careful weighing of the benefits and drawbacks of
the various options – “can encourage research that may some day issue
in a set theory that is clearly best.” As we might expect, his attitude
here is much like the attitude one could have with regard to the progress
of science generally. Various theories are developed until one emerges as
best, however temporary this privileged position might happen to be.
This careful comparative development of set theory with regard to its
existence assumptions and axiomatic strength is, for Quine, the philoso-
phy of set theory. This should come as no surprise to anyone who is
familiar with his approach to the philosophy of the sciences generally.
Neither in the natural sciences nor in mathematics do we look for a first
philosophy. Our starting point is within science itself, so in the case of
set theory, its philosophy begins within set theory rather than in some
extramathematical metaphysical conception to which it must be fitted.

Ibid., p. .

Ibid., p. .

Something like this seems to be the source of the dispute that arose between Quine and Donald
Martin over the latter’s review of Set Theory and Its Logic. See Martin’s “Review of Set Theory and
Its Logic,” Journal of Philosophy : (), pp. –, and Quine’s “Reply to Martin,” Journal
of Philosophy : (), pp. –.

Set Theory and Its Logic, p. x.
 New Foundations, the Axiom of Choice, and Arithmetic 
Having cleared his readers’ minds of set-theoretic prejudice in the first
two parts of Set Theory and Its Logic, Quine considers in the third part
the familiar set theories of Russell, Zermelo, and himself. Rather than
drawing distinctions and building boundaries between them, Quine takes
the very radical approach of bringing these various and often incompati-
ble set theories into dialog with each other. So let us turn to the details
of Quine’s picture and how it might further the growth and knowledge
of set theory.
Quine begins his study with Russell’s theory of types, progressing next
to Zermelo’s theory and finally to his own NF. This account is not his-
torical; nor is it intended to be. Rather, we can view Quine’s account as
a sort of logical progression through the various set theories that he con-
siders, showing how each one can be seen, in a sense, as emerging from
the other. Boolos and Russell, among others, argue against Quine’s NF
by remarking on its seemingly artificial development as only a response
to the paradoxes, asserting that there is no motivating thought behind it
other than this. Quine largely accepts this but sees it as the situation gen-
erally in set theory after the paradoxes. To him, it is not a criticism of
NF any more than it is a criticism of the other set theories that he con-
siders. The paradoxes force such artificiality upon all of us. As we have
heard over and over from Quine, our most intuitive thought about sets,
that is, as extensions of predicates, led us into contradiction, and we have
seen this play out in the early history of set theory. Talk of sets existing
in some sort of hierarchy did not enter in as an intuitive first thought
about sets but rather emerged from considerations of paradoxes, first in
Russell’s work and sometime later in Zermelo’s (much later than his first
axiomatization). Artificial development, then, is unavoidable.
Perhaps critics of NF would grant Quine’s point that all set theory is in
part artificial, but for them, NF is just a little too artificial in comparison
to the theory of types or Zermelo’s set theory. Quine’s comparative
account of set theory can be read as a response to just this sort of criti-
cism. Aside from comparing the relative merits of the various set theories
and how their respective disadvantages may lead one to a different axioma-
tization, Quine – perhaps even more importantly – brings out what uni-
fies them. To this end, he begins with types and shows how its
hierarchical structure leads to Zermelo’s theory and then finally to his own
NF. What Quine does is take this talk of hierarchy and abstract away


I should include here Martin’s “Review of Set Theory and Its Logic.” See also Quine’s “Reply to
Martin.”
 Quine, New Foundations, and the Philosophy of Set Theory
from this metaphor as much as possible until he is left with its barest logi-
cal structure, present across the boundaries of the various competing and
incompatible axiomatizations of set theory. This comparison accounts for
certain intuitions about set theory while also moving against our becoming
too attached to certain intuitions that we might have about set theory.
The final section of Set Theory and Its Logic, then, is a crucial move in an
argument against those who would focus on just one formulation of set
theory “to the point of retraining our intuition to it.”
Let us see now how this argument goes. We discussed in Chapter 
how Russell saw his theory of types as emerging from Henri Poincaré’s
diagnosis that the contradictions all have their origins in quantifying over
illegitimate totalities, which Russell took to be summed up in his vicious
circle principle. Recall that in its simplified version, the theory of types
divides the universe into levels: individuals at the lowest level, then
classes of individuals, then classes of classes of individuals, and so on. For
something to be a member of a class, it must be of level n while the class
of which it is a member is of level n + . This dividing of the universe
into levels, motivated by Russell’s vicious circle principle, apparently
restored consistency to the theory of classes. But Quine observes that
this move is drastic in that its type restrictions tamper with the original
logic. Furthermore (as we saw in Chapters  and ), type theory has the
undesirable feature of reduplicating objects in other levels of the hierar-
chy, among them the empty class, a series of quasi-universal classes, and
the various objects of mathematics such as the different number systems.
We might take these drawbacks alone as moving us beyond types to
some other approach to set theory, but Quine observes that there is also
something valuable in the constructive metaphor of type theory: “it is a
part of set theory that carries extra conviction, because of the construction
metaphor.” Quine then moves to Zermelo set theory with its unrestricted
(with regard to type) variables not merely as a reaction to the undesirable
features of types but rather as a natural generalization of types’ hierarchical
structure. Whereas other philosophers of set theory have ruled out types as
set theory, often by fiat, Quine engages with both theories to demonstrate
how we might see the interconnections between them. Let us consider

Set Theory and Its Logic, p. .

Ibid., p. .

The standard move is to declare types to be part of logic, that is, higher-order logic, and so not of
set theory. Quine remarks on this in Set Theory and Its Logic, pp. –, and in many other
places in his writings. Boolos makes this sort of declaration in his  “On Second-Order
Logic,” in Logic, Logic, and Logic, p. , fn. , in discussing Quine’s presentation of a class theory
in the third edition of his Methods of Logic (Chicago, IL: Holt Rinehart, and Winston, ).
 New Foundations, the Axiom of Choice, and Arithmetic 
Quine’s approach by again looking at how Zermelo’s set theory emerges in
a natural way from Russell’s theory of types.
To show how Zermelo’s theory emerges from Russell’s, Quine first
observes that typical ambiguity in types does not change a many-sorted
theory into one with general variables. The point of typical ambiguity is
that although we often do not explicitly specify the types involved for
each formula (for ease of writing and reading), we must still be certain
that they can be made explicit in the appropriate way if we are called
upon to do so. So, for example, the typically ambiguous formula
“(∃y)(∀x)(x ∈ y)” can be shown to be legitimate by restoring type indexes
according to the scheme “(∃yn+)(∀xn)(xn ∈ yn+).” However, Quine
observes that we may also have reasons to consider the translation of the
many-sorted types into a theory of general variables. For instance, the
type indices are themselves cumbersome, but typical ambiguity can make
things worse in some cases. Consider the two formulas “x ∈ y” and
“y ∈ x.” Both are meaningful under types, yet “x ∈ y . y ∈ x” is not.
In this way, we see that the theory of types blocks the set-theoretic con-
tradictions with the drastic move of revising general logic. Quine, favor-
ing the maxim of minimum mutilation, thinks that this is to be avoided
where possible. Furthermore, given his comparative interest in set theory
(and scientific theories generally), a single underlying logic makes such
studies far more feasible.
Quine argues that the many-sorted logic is not essential to type the-
ory. Instead, we can allow the variables to be completely general and
add a predicate Tn to impose the type restrictions. Formulas can then
be restricted in the familiar way as follows: “(∀x)(Tnx ⊃ Fx)” or
“(∃x)(Tnx . Fx).” This move, Quine says, makes sense of Russell’s gram-
matical restrictions. Whereas Russell declared all formulas of the form
“xm ∈ yn” with n ≠ m +  meaningless, we can now take such formulas
to be simply false. Therefore the grammatical restrictions generated by
types appear unnecessary in the first place. The two axiom schemas of
types go over into the new theory as follows: Comprehension and
extensionality become “(∃y)(Tn+y . (∀x)(Tnx ⊃ x ∈ y ≡ Fx)” and
“(Tn+x . Tn+y . (∀w)(Tnw ⊃ (w ∈ x ≡ w ∈ y))) ⊃ x = y,” respectively.
(Officially, identity is defined in terms of membership and first-order
logic and so is eliminable.)


This was one of Quine’s earliest motivations for his attempts to rework Russell’s logic. See
Quine’s remarks on his  “Set-Theoretic Foundations for Logic,” in his “The Inception of
New Foundations” in Selected Logic Papers, pp. –.
 Quine, New Foundations, and the Philosophy of Set Theory
Having shown the eliminability of typed variables in favor of typing
predicates, Quine goes on to show how smoothly types can lead us into
Zermelo’s set theory by converting the restricted variables of types into
cumulative types with general variables. We begin by equating the
empty sets of all the various types so as to get a single empty set. Quine
then introduces the trick of identifying individuals, the memberless
objects of type , with their unit classes and treating membership
between individuals as identity. The objects of type , the Tx’s – the
individuals – can now be defined by the formula “(∀y)(y ∈ x ≡ y = x)”
and the objects of higher type, the Tn+x’s – the classes – by the
formula “(∀y)(y ∈ x ⊃ Tny).” Any typed variable can be rewritten
according to these definitions so that we use only completely general
variables, and the types become cumulative, that is, Tmx ⊃ Tnx for
all m < n.
Here, we see just the sort of philosophy of set theory in which Quine
engages. Rather than looking at differences and attempting to privilege
one set theory over another, he tries to see their similarities. For Quine,
coming to understand the realm of sets means investigating sets from the
various perspectives that different set theories allow for. His endeavor in
set theory, as in much of science, is cooperative rather than exclusionary
and aims to broaden our knowledge through pluralism about set theory.
This moving from noncumulative to cumulative types perhaps allows us
to better understand the apparent hierarchical structure of sets. It was
always present in both types and Zermelo set theory, and Quine has
now shown us explicitly how the idea connects the two approaches to set
theory. Quine himself found this connection immensely striking. He
later observed of Zermelo’s theory that “in its multiplicity of axioms it
seemed inelegant, artificial, and ad hoc. I had not yet appreciated how
naturally his system emerges from the theory of types when we render


The details can be found in Set Theory and Its Logic, pp. –. Here, I hope to give only the
basic idea behind the conversion to cumulative types and Zermelo set theory. This was an idea
that Quine was already onto much earlier, which he laid out in his “Unification of Universes in
Set Theory,” Journal of Symbolic Logic : (), pp. –. It is worth noting that Quine
here sees types and Zermelo’s theory as sharing more with each other than with NF, and although
he states a preference for his own theory, he also observes that “in the present paper I am pursuing
other lines in order to dramatize the process of unifying the universes of theories containing multi-
ple styles of variables” (p. ). Here, we see the distinct approach that Quine takes to set theory.
While so much of the philosophy of set theory concerns singling out a privileged theory, Quine
instead looks for where they might be of a piece with one another.

Individuals are necessary to types in accounting for the infinite. The axiom of infinity for types
says that there are an infinite number of individuals.
 New Foundations, the Axiom of Choice, and Arithmetic 
the types cumulative and describe them by means of general variables.”
So while many have remarked on the artificiality of Quine’s systems,
Quine initially thought the same of Zermelo’s. The sort of elegance, sim-
plicity, and unity that Quine (like many other philosophers of science)
has so often put forth as desirable for scientific theories generally comes
now also to Zermelo’s theory by way of its emergence from the perhaps
more intuitive theory of types. There are, of course, a variety of ways in
which we might consider types to be intuitive. Here, I have in mind (as
I suspect Quine does) that types, at least initially, seemed to do the least
damage to the original conception of sets as extensional and specified by
a comprehension principle of a sort. In linking these theories, this sort of
intuitiveness of types passes on to Zermelo. Indeed, Quine concludes
that had he appreciated this link earlier, “I might not have pressed on to
‘New Foundations.’”
Bringing out such connections now also gives us another way to see
NF as sharing an important aspect with types and Zermelo set theory.
The move from types to Zermelo’s theory perhaps gives us further reason
to consider the idea of a hierarchy as somehow essential to what a set is.
But as we saw in Chapter , it is notoriously difficult to make philoso-
phical sense of this idea. On the one hand, we have talk of sets coming
into being at particular levels of the hierarchy; on the other hand, we
have talk of sets existing eternally from the past and to the future. With
NF, Quine perhaps provides a way of making sense of this idea of hierar-
chy while freeing it from a particular metaphysical account of sets in the
following way. We can think of NF as abstracting away completely from
the idea of a hierarchy until we are left with a much weaker purely syn-
tactic account of set existence. As Burton Dreben has put it – and
Joseph Ullian, following him – Quine’s approach is one of syntactic
exploration. The hierarchy gives us only the numbering scheme that is
used as NF’s syntactic test for set existence, an idea that Quine came
upon by taking typical ambiguity very seriously, which appears to have
no need for an actual hierarchy of sets. We need not make the further


Quine, “The Inception of New Foundations” (), in Selected Logic Papers, enlarged edn
(Cambridge, MA: Harvard University Press ), p. .

See, for example, Quine’s “Posits and Reality,” in The Ways of Paradox and Other Essays, pp. –.

Quine, “The Inception of New Foundations,” p. .

Ullian, “Quine and the Field of Mathematical Logic,” pp.  and , n. .

Quine, Set Theory and Its Logic, pp. –. See also his remarks in “The Inception of New
Foundations,” p. , where he discusses how he arrived at NF by considering Russell’s types in
conjunction with Zermelo’s theory.
 Quine, New Foundations, and the Philosophy of Set Theory
leap to, say, the iterative hierarchy to get an account of the essence of set-
hood. This would be an extra, more robust philosophical assumption.
Quine’s syntactic account instead keeps our feet on the ground. In
doing the philosophy of set theory, Quine tries to keep to his naturalistic
strictures, and we must always remember that for him, “philosophy of
science is philosophy enough.”


Quine makes this remark in his Methods of Logic, th edn, p. , when considering the substitu-
tional versus set-theoretic account of logical truth. The point seems apt here as well: We try to get
by with fewer assumptions where possible.

W. V. Quine, “Mr. Strawson on Logical Theory,” in The Ways of Paradox and Other Essays,
p. .
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 Index

absolute infinite, , , , , , , , Burali-Forti paradox, , , , , , –
,  Burali-Forti, Cesare, , , , , 
actual infinite, , , , , , 
aleph,  calculus
system of all, , – rigorization of, –
aleph notation,  Cantor, Georg
analytic/synthetic distinction, , , , , actual infinite, –, , 
, , ,  cardinal comparability, , 
axiom of choice, , , , , , , , cardinal numbers, –
, , –,  consistent and inconsistent multiplicities,
Banach-Tarski paradox, , , ,  –
epistemological status of, ,  construction of real numbers, , –
inconsistent with axiom of determinacy, continuum hypothesis, , –
– diagonal method, –
independent of Z and ZF,  numbering, –
axiom of counting, –,  order types, , 
axiom of determinacy, , –,  ordinal comparability, 
axiom of extensionality, , , , , –, ordinal numbers, , , , –
,  power of a set, , , , , , , 
axiom of inf,  principles of generation, –
axiom of infinity, , , , , , , rigorization of calculus, 
, , , , ,  set-theoretic paradoxes, , , 
axiom of regularity, , , –,  transfinite numbers, –, , 
axiom of replacement, , –,  uncountable infinite, –, , 
well-ordering, , –
big sets, , , , –, , , , Cantor’s paradox, , , , , 
,  Cantor’s theorem, , , 
eliminated, , ,  Cantor’s paradox, , , , , –, 
as source of paradoxes,  Cantor’s theorem, , –
Boolos, George,  cardinals
ideal theory, , ,  cardinal numbers, 
interative conception of set,  Carnap, Rudolf, , 
iterative conception of set, –,  against metaphysics, , , , ,
on axiom of choice, – , 
on axiom of extensionality, – explication, , , , 
on axiom of regularity, – internal/external distinction, –, 
on axiom of replacement, – linguistic frameworks, , 
on Cantor’s conception of set, – principle of tolerance, , –,
on higher-order logic,  , 
on naïve set theory,  Cauchy, Agustin-Louis, , , , 
on NF, , ,  comprehension axiom, 


 Index
comprehension principle, , , , , , , Lagrange, Joseph-Louis, , 
, , , , , , , large cardinals, , , 
, , , –, ,  Leibniz, Gottfried Wilhelm, , , 
Linnebo, Øystein, 
Dedekind, Richard, , ,  Linsky, Bernard, 
Dirichlet, Peter Gustav Lejeune, 
Dreben, Burton,  Martin, Donald, , 
Mendelson, Elliot, 
Ebbinghaus, Heinz-Dieter,  Mirimanoff, Dimitri, 
Euler, Leonhard, ,  Moore, George Edward, , , 
explication,  Moore, Gregory, , , , , , , 

Feferman, Solomon,  New Foundations

Fourier, Joseph,  and axiom of choice, 
Fraenkel, Abraham, ,  axiom of choice disproved in, , , 
Frege, Gottlob, , , , ,  axiom of choice restricted in, –, ,
, 
Garciadiego, Alejandro, ,  axioms of, 
Gentzen, Gerhard, , , ,  Cantorian sets, 
Gödel, Kurt, , , , , , , , , Cantor’s theorem in, 
, , ,  consistency of, 
and explication, 
Hallett, Michael, , , ,  as framework for mathematics, , 
Hart, W. D., , , ,  generalized continuum hypothesis disproved
Hegel, Georg Wilhelm Friedrich, ,  in, 
Herbrand, Jacques,  induction in, 
higher-order logic, , , ,  iterative conception and, , , 
Hilbert, David, , ,  non-Cantorian sets, 
Hilbert’s program, , ,  pragmatic justification of, 
Hylton, Peter, , , , , , , stratification in, 
,  theory of types and, , , , –
Zermelo’s theory and, , , 
incompleteness theorems, , , , , ,  Newton, Sir Isaac, , 
iterative conception of set, , , , , , non-well-founded set theory, , , , ,
,  , , 
and set theory as conceptual analysis, , ,
,  ordered-pair, 
consistency of, 
pragmatic justification of, ,  Parsons, Charles, , 
role of intuitions in, , –, , Peano, Giuseppe, , , 
,  potential infinite, , , , 

Jaskowski, Stanislaw,  Quine, W. V.

Jech, Thomas, , –,  against metaphysics, , 
Jensen, Ronald,  axiom of choice, 
John Barwise,  axiom of reducibility, –
calrification of conceptual scheme, 
Kanamori, Akihiro, , , , , , canonical notation, 
,  clarification of conceptual scheme, ,
Kant, Immanuel, ,  –, 
Kleinberg, Eugene M.,  consistency proofs, , –
König, Julius,  convention, –
Kreisel, Georg, –,  explication, , –
Kripke, Saul,  extensionality, –
Kuratowski, Kazimierz, ,  higher-order logic, , , 
 Index 
holism,  intensional basis for logic, –
infinte sets,  large classes, , 
intutions in set theory, – nature of logic, 
iterative conception of set,  pure mathematics, 
logic, , , – set theory as explication, , 
Logic of Sequences, – theory of types, –
natural deduction,  vicious circle principle, 
naturalism,  Russell’s paradox, , –, , , ,
ontological commitment, – , 
ontological parsimony, –
ontology,  Scott, Dana, , 
set theory as comparative, , – semantic paradoxes, , , 
set theory as explication, , , – set
set theory as exploratory,  extension of a concept or predicate, , , ,
theory of types, –,  , , , , , , ,
tolerance, –,  , 
virtual theory of classes, – extension of concept or predicate, 
Zermelo’s theory and, , – set theory as conceptual analysis, , , 
set theory as explication, , , , 
Ramsey, F. P., ,  Shoenfield, Joseph, 
Resnik, Michael, – Specker, Ernst, 
Rieger, Adam, , 
Riemann, Berhard,  Ullian, Joseph, , 
Riemann, Bernhard, 
Robinson, Abraham,  von Neumann, John, , 
Rosser, J. Barkley, , , –, 
Russell, Bertrand Wang, Hao, , , , 
against Idealism, , – Weierstrass, Karl, , 
analysis, , – Wiener, Norbert, , 
axiom of reducibility, –
cardinal numbers,  Zermelo, Ernst, , 
Hegelianism,  axiomatization of set theory, –
idealism of,  set theory as explication, , , 
inductive, or regressive, justification, – well-ordering theorem, 