Stanford Encyclopedia of

Philosophy
The Early
Development of
Set Theory
First published Tue Apr 10, 2007;
substantive revision Thu Jun 18, 2020

Set theory is one of the greatest achievements of modern

mathematics. Basically all mathematical concepts, methods,
and results admit of representation within axiomatic set
theory. Thus set theory has served quite a unique role by
systematizing modern mathematics, and approaching in a
unified form all basic questions about admissible
mathematical arguments—including the thorny question of
existence principles. This entry covers in outline the
convoluted process by which set theory came into being,
covering roughly the years 1850 to 1930.

In 1910, Hilbert wrote that set theory is

that mathematical discipline which today occupies an

outstanding role in our science, and radiates
[ausströmt] its powerful influence into all branches of
mathematics. [Hilbert 1910, 466; translation by entry
author]
This already suggests that, in order to discuss the early
history, it is necessary to distinguish two aspects of set
theory: its role as a fundamental language and repository of
the basic principles of modern mathematics; and its role as
an independent branch of mathematics, classified (today) as
a branch of mathematical logic. Both aspects are considered
here.

The first section examines the origins and emergence of set

theoretic mathematics around 1870; this is followed by a
discussion of the period of expansion and consolidation of
the theory up to 1900. Section 3 provides a look at the
critical period in the decades 1897 to 1918, and Section 4
deals with the time from Zermelo to Gödel (from theory to
metatheory), with special attention to the often overlooked,
but crucial, descriptive set theory.

1. Emergence
2. Consolidation
3. Critical Period
4. From Zermelo to Gödel
Bibliography
Cited Works
Further Reading
Academic Tools
Other Internet Resources
Related Entries

1. Emergence
The concept of a set appears deceivingly simple, at least to
the trained mathematician, and to such an extent that it
becomes difficult to judge and appreciate correctly the
contributions of the pioneers. What cost them much effort
to produce, and took the mathematical community
considerable time to accept, may seem to us rather self-
explanatory or even trivial. Three historical misconceptions
that are widespread in the literature should be noted at the
outset:

1. It is not the case that actual infinity was universally

rejected before Cantor.
2. Set-theoretic views did not arise exclusively from analysis,
but emerged also in algebra, number theory, and
geometry.
3. In fact, the rise of set-theoretic mathematics preceded
Cantor’s crucial contributions.

All of these points shall become clear in what follows.

The notion of a collection is as old as counting, and logical

ideas about classes have existed since at least the “tree of
Porphyry” (3rd century CE). Thus it becomes difficult to sort
out the origins of the concept of set. But sets are neither
collections in the everyday sense of this word, nor
“classes” in the sense of logicians before the mid-19th
century. The key missing element is objecthood — a set is a
mathematical object, to be operated upon just like any other
object (the set N is as much ‘a thing’ as number 3). To
clarify this point, Russell employed the useful distinction
between a class-as-many (this is the traditional idea) and a
class-as-one (or set).

Ernst Zermelo, a crucial figure in our story, said that the

theory had historically been “created by Cantor and
Dedekind” [Zermelo 1908, 262]. This suggests a good
pragmatic criterion: one should start from authors who have
significantly influenced the conceptions of Cantor, Dedekind,
and Zermelo. For the most part, this is the criterion adopted
here. Nevertheless, as every rule calls for an exception, the
case of Bolzano is important and instructive, even though
Bolzano did not significantly influence later writers.

In 19th century German-speaking areas, there were some

intellectual tendencies that promoted the acceptance of the
actual infinite (e.g., a revival of Leibniz’s thought). In spite
of Gauss’s warning that the infinite can only be a manner of
speaking, some minor figures and three major ones
(Bolzano, Riemann, Dedekind) preceded Cantor in fully
accepting the actual infinite in mathematics. Those three
authors were active in promoting a set-theoretic formulation
of mathematical ideas, with Dedekind’s contribution in a
good number of classic writings (1871, 1872, 1876/77, 1888)
being of central importance.

Chronologically, Bernard Bolzano was the first, but he

exerted almost no influence. The high quality of his work in
logic and the foundations of mathematics is well known. A
book entitled Paradoxien des Unendlichen was
posthumously published in 1851. Here Bolzano argued in
detail that a host of paradoxes surrounding infinity are
logically harmless, and mounted a forceful defence of actual
infinity. He proposed an interesting argument attempting to
prove the existence of infinite sets, which bears comparison
with Dedekind’s later argument (1888). Although he
employed complicated distinctions of different kinds of sets
or classes, Bolzano recognized clearly the possibility of
putting two infinite sets in one-to-one correspondence, as
one can easily do, e.g., with the intervals [0, 5] and [0, 12] by
the function 5y = 12x. However, Bolzano resisted the
conclusion that both sets are “equal with respect to the
multiplicity of their parts” [1851, 30–31]. In all likelihood,
traditional ideas of measurement were still too powerful in
his way of thinking, and thus he missed the discovery of the
concept of cardinality (however, one may consider Non-
Cantorian ideas, on which see Mancosu 2009).

The case of Bolzano suggests that a liberation from metric

concepts (which came with the development of theories of
projective geometry and especially of topology) was to have
a crucial role in making possible the abstract viewpoint of set
theory. Bernhard Riemann proposed visionary ideas about
topology, and about basing all of mathematics on the notion
of set or “manifold” in the sense of class
(Mannigfaltigkeit), in his celebrated inaugural lecture “On
the Hypotheses which lie at the Foundations of Geometry”
(1854/1868a). Also characteristic of Riemann was a great
emphasis on conceptual mathematics, particularly visible in
his approach to complex analysis (which again went deep
into topology). To give but the simplest example, Riemann
was an enthusiastic follower of Dirichlet’s idea that a
function has to be conceived as an arbitrary correspondence
between numerical values, be it representable by a formula
or not; this meant leaving behind the times when a function
was defined to be an analytic expression. Through this new
style of mathematics, and through his vision of a new role for
sets and a full program for developing topology, Riemann
was a crucial influence on both Dedekind and Cantor (see
Ferreirós 1999).

The five-year period 1868–1872 saw a mushrooming of set-

theoretic proposals in Germany, so much so that we could
regard it as the birth of set-theoretic mathematics.
Riemann’s geometry lecture, delivered in 1854, was
published by Dedekind in 1868, jointly with Riemann’s
paper on trigonometric series (1854/1868b, which presented
the Riemann integral). The latter was the starting point for
deep work in real analysis, commencing the study of
“seriously” discontinuous functions. The young Georg
Cantor entered into this area, which led him to the study of
point-sets. In 1872 Cantor introduced an operation upon
point sets (see below) and soon he was ruminating about
the possibility to iterate that operation to infinity and
beyond: it was the first glimpse of the transfinite realm.

Meanwhile, another major development had been put

forward by Richard Dedekind in 1871. In the context of his
work on algebraic number theory, Dedekind introduced an
essentially set-theoretic viewpoint, defining fields and ideals
of algebraic numbers. These ideas were presented in a very
mature form, making use of set operations and of structure-
preserving mappings (see a relevant passage in Ferreirós
1999: 92–93; Cantor employed Dedekind’s terminology for
the operations in his own work on set theory around 1880
[1999: 204]). Considering the ring of integers in a given field
of algebraic numbers, Dedekind defined certain subsets
called “ideals” and operated on these sets as new objects.
This procedure was the key to his general approach to the
topic. In other works, he dealt very clearly and precisely with
equivalence relations, partition sets, homomorphisms, and
automorphisms (on the history of equivalence relations, see
Asghari 2‎ 018). Thus, many of the usual set-theoretic
procedures of twentieth-century mathematics go back to his
work. Several years later (in 1888), Dedekind would publish a
presentation of the basic elements of set theory, making only
a bit more explicit the operations on sets and mappings he
had been using since 1871.
The following year, Dedekind published a paper [1872] in
which he provided an axiomatic analysis of the structure of
the set R of real numbers. He defined it as an ordered field
that is also complete (in the sense that all Dedekind-cuts on
R correspond to an element in R); completeness in that

sense has the Archimedean axiom as a consequence. Cantor

too provided a definition of R in 1872, employing Cauchy
sequences of rational numbers, which was an elegant
simplification of the definition offered by Carl Weierstrass in
his lectures. The form of completeness axiom that
Weierstrass preferred was Bolzano’s principle that a
sequence of nested closed intervals in R (a sequence such
that [a
m+1 ,b
m+1 ] ⊂ [a , b ]) “contains” at least one real
m m

number (or, as we would say, has a non-empty intersection).

The Cantor and Dedekind definitions of the real numbers

relied implicitly on set theory, and can be seen in retrospect
to involve the assumption of a Power Set principle. Both took
as given the set of rational numbers, and for the definition of
R they relied on a certain totality of infinite sets of rational

numbers (either the totality of Cauchy sequences, or of all

Dedekind cuts). With this, too, constructivistic criticism of set
theory began to emerge, as Leopold Kronecker started to
make objections to such infinitary procedures.
Simultaneously, there began a study of the topology of R, in
particular in the work of Weierstrass, Dedekind, and Cantor.
The set-theoretic approach was also exploited by several
authors in the fields of real analysis and complex analysis
(e.g., Hankel, du Bois-Reymond, H.J.S. Smith, U. Dini) and by
Dedekind in joint work with Weber (1882), pioneering
algebraic geometry.

Cantor’s derived sets are of particular interest (for the

context of this idea in real analysis, see e.g., Dauben 1979,
Hallett 1984, Lavine 1994, Kanamori 1996, Ferreirós 1999).
Cantor took as given the “conceptual sphere” of the real
numbers, and he considered arbitrary subsets P , which he
called ‘point sets’. A real number r is called a limit point of
P , when all neighbourhoods of r contain points of P . This

can only happen if P is infinite. With that concept, due to

Weierstrass, Cantor went on to define the derived set P of P ′

, as the set of all the limit points of P . In general P may be

′

infinite and have its own limit points (see Cantor’s paper in
Ewald [1996, vol. 2, 840ff], esp. p. 848). Thus one can iterate
the operation and obtain further derived sets P , P … P
′′ ′′′ (n)

… It is easy to give examples of a set P that will give rise to

non-empty derived sets P for all finite n. (A rather trivial
(n)

example is P = Q , the set of rational numbers in the unit

[0,1]

interval; in this case P = [0, 1] = P .) Thus one can define

′ ′′

P
(∞)
as the intersection of all P for finite n. This was
(n)

Cantor’s first encounter with transfinite iterations.

Then, in late 1873, came a surprising discovery that fully

opened the realm of the transfinite. In correspondence with
Dedekind (see Ewald 1996, vol. 2), Cantor asked the question
whether the infinite sets N of the natural numbers and R of
real numbers can be placed in one-to-one correspondence.
In reply, Dedekind offered a surprising proof that the set A of
all algebraic numbers is denumerable (i.e., there is a one-to-
one correspondence with N). A few days later, Cantor was
able to prove that the assumption that R is denumerable
leads to a contradiction. To this end, he employed the
Bolzano-Weierstrass principle of completeness mentioned
above. Thus he had shown that there are more elements in
R than in N or Q or A, in the precise sense that the

cardinality of R is strictly greater than that of N.

2. Consolidation
Set theory was beginning to become an essential ingredient
of the new “modern” approach to mathematics. But this
viewpoint was contested, and its consolidation took a rather
long time. Dedekind’s algebraic style only began to find
followers in the 1890s; David Hilbert was among them. The
soil was better prepared for the modern theories of real
functions: Italian, German, French and British mathematicians
contributed during the 1880s. And the new foundational
views were taken up by Peano and his followers, by Frege to
some extent, by Hilbert in the 1890s, and later by Russell.

Meanwhile, Cantor spent the years 1878 to 1885 publishing

key works that helped turn set theory into an autonomous
branch of mathematics. Let’s write A ≡ B in order to
express that the two sets A, B can be put in one-to-one
correspondence (have the same cardinality). After proving
that the irrational numbers can be put in one-to-one
correspondence with R, and, surprisingly, that also R ≡ R,
n

Cantor conjectured in 1878 that any subset of R would be

either denumerable (≡ N) or ≡ R. This is the first and
weakest form of the celebrated Continuum Hypothesis.
During the following years, Cantor explored the world of
point sets, introducing several important topological ideas
(e.g., perfect set, closed set, isolated set), and arrived at
results such as the Cantor-Bendixson theorem.

A point set P is closed iff its derived set P ⊆ P , and perfect

′

iff P = P . The Cantor-Bendixson theorem then states that a

′

closed point set can be decomposed into two subsets R and

S , such that R is denumerable and S is perfect (indeed, S is

the ath derived set of P , for a countable ordinal a). Because

of this, closed sets are said to have the perfect set property.
Furthermore, Cantor was able to prove that perfect sets have
the power of the continuum (1884). Both results implied that
the Continuum Hypothesis is valid for all closed point sets.
Many years later, in 1916, Pavel Aleksandrov and Felix
Hausdorff were able to show that the broader class of Borel
sets have the perfect set property too.

His work on points sets led Cantor, in 1882, to conceive of

the transfinite numbers (see Ferreirós 1999: 267ff). This was a
turning point in his research, for from then onward he
studied abstract set theory independently of more specific
questions having to do with point sets and their topology
(until the mid-1880s, these questions had been prominent in
his agenda). Subsequently, Cantor focused on the transfinite
cardinal and ordinal numbers, and on general order types,
independently of the topological properties of R.

The transfinite ordinals were introduced as new numbers in

an important mathematico-philosophical paper of 1883,
Grundlagen einer allgemeinen Mannigfaltigkeitslehre (notice
that Cantor still uses Riemann’s term Mannigfaltigkeit or
‘manifold’ to denote sets). Cantor defined them by means
of two “generating principles”: the first (1) yields the
successor a + 1 for any given number a, while the second (2)
stipulates that there is a number b which follows immediately
after any given sequence of numbers without a last element.
Thus, after all the finite numbers comes, by (2), the first
transfinite number, ω (read: omega); and this is followed by
ω + 1, ω + 2, …, ω + ω = ω ⋅ 2, …, ω ⋅ n, ω ⋅ n + 1, …, ω , ω + 1, …,
2 2

ω , … and so on and on. Whenever a sequence without last

ω

element appears, one can go on and, so to say, jump to a

higher stage by (2).
The introduction of these new numbers seemed like idle
speculation to most of his contemporaries, but for Cantor
they served two very important functions. To this end, he
classified the transfinite ordinals as follows: the “first
number class” consisted of the finite ordinals, the set N of
natural numbers; the “second number class” was formed
by ω and all numbers following it (including ω , and many
ω

more) that have only a denumerable set of predecessors.

This crucial condition was suggested by the problem of
proving the Cantor-Bendixson theorem (see Ferreirós 1995).
On that basis, Cantor could establish the results that the
cardinality of the “second number class” is greater than
that of N; and that no intermediate cardinality exists. Thus, if
you write card(N) = ℵ (read: aleph zero), his theorems
0

justified calling the cardinality of the “second number

class” ℵ .
1

After the second number class comes a “third number

class” (all transfinite ordinals whose set of predecessors has
cardinality ℵ ); the cardinality of this new number class can
1

be proved to be ℵ . And so on. The first function of the

2

transfinite ordinals was, thus, to establish a well-defined

scale of increasing transfinite cardinalities. (The aleph
notation used above was introduced by Cantor only in 1895.)
This made it possible to formulate much more precisely the
problem of the continuum; Cantor’s conjecture became the
hypothesis that card(R) = ℵ . Furthermore, relying on the
1

transfinite ordinals, Cantor was able to prove the Cantor-

Bendixson theorem, rounding out the results on point sets
that he had been elaborating during these crucial years. The
Cantor-Bendixson theorem states: closed sets of R n

(generalizable to Polish spaces) have the perfect set

property, so that any closed set S in R can be written
n
uniquely as the disjoint union of a perfect set P and a
countable set R. Moreover, P is S for α countable ordinal.
α

The study of the transfinite ordinals directed Cantor’s

attention towards ordered sets, and in particular well-
ordered sets. A set S is well-ordered by a relation < iff < is a
total order and every subset of S has a least element in the
<-ordering. (The real numbers are not well-ordered in their
usual order: just consider an open interval. Meanwhile, N is
the simplest infinite well-ordered set.) Cantor argued that
the transfinite ordinals truly deserve the name of numbers,
because they express the “type of order” of any possible
well-ordered set. Notice also that it was easy for Cantor to
indicate how to reorder the natural numbers so as to make
them correspond to the order types ω + 1, ω + 2, …, ω ⋅ 2, …,
ω ⋅ n, …, ω , …, ω , … and so on. (For instance, reordering N in
2 ω

the form: 2, 4, 6, …, 5, 15, 25, 35, …, 1, 3, 7, 9, … we obtain a

set that has order type ω ⋅ 3.)

Notice too that the Continuum Hypothesis, if true, would

entail that the set R of real numbers can indeed be well-
ordered. Cantor was so committed to this viewpoint, that he
presented the further hypothesis that every set can be well-
ordered as “a fundamental and momentous law of
thought”. Some years later, Hilbert called attention to both
the Continuum Hypothesis and the well-ordering problem as
Problem 1 in his celebrated list of ‘Mathematische
Probleme’ (1900). Doing so was an intelligent way of
emphasizing the importance of set theory for the future of
mathematics, and the fruitfulness of its new methods and
problems.

In 1895 and 1897, Cantor published his last two articles. They
were a well-organized presentation of his results on the
transfinite numbers (cardinals and ordinals) and their theory,
and also on order types and well-ordered sets. However,
these papers did not advance significant new ideas.
Unfortunately, Cantor had doubts about a third part he had
prepared, which would have discussed very important issues
having to do with the problem of well-ordering and the
paradoxes (see below). Surprisingly, Cantor also failed to
include in the 1895/97 papers a theorem which he had
published some years before which is known simply as
Cantor’s Theorem: given any set S , there exists another set
whose cardinality is greater (this is the power set P(S), as we
now say—Cantor used instead the set of all functions of the
form f : S → {0, 1}, which is equivalent). In the same short
paper (1892), Cantor presented his famous proof that R is
non-denumerable by the method of diagonalisation, a
method which he then extended to prove Cantor’s
Theorem. (A related form of argument had appeared earlier
in the work of P. du Bois-Reymond [1875], see among others
[Wang 1974, 570] and [Borel 1898], Note II.)

Meanwhile, other authors were exploring the possibilities

opened by set theory for the foundations of mathematics.
Most important was Dedekind’s contribution (1888) with a
deep presentation of the theory of the natural numbers. He
formulated some basic principles of set (and mapping)
theory; gave axioms for the natural number system; proved
that mathematical induction is conclusive and recursive
definitions are flawless; developed the basic theory of
arithmetic; introduced the finite cardinals; and proved that
his axiom system is categorical. His system had four axioms.
Given a function φ defined on S , a set N ⊆ S , and a
distinguished element 1 ∈ N , they are as follows:
(α) ϕ(N ) ⊂ N

(β) N = ϕo {1}

(γ) 1 ∉ ϕ(N )

(δ) the function ϕ is injective.

Condition (β) is crucial since it ensures minimality for the set

of natural numbers, which accounts for the validity of proofs
by mathematical induction. N = ϕ {1} is read: N is the chain
o

of singleton {1} under the function φ, that is, the minimal

closure of {1} under the function φ. In general, one considers
the chain of a set A under an arbitrary mapping γ, denoted
by γ (A); in his booklet Dedekind developed an interesting
o

theory of such chains, which allowed him to prove the

Cantor-Bernstein theorem. The theory was later generalized
by Zermelo and applied by Skolem, Kuratowski, etc.

In the following years, Giuseppe Peano gave a more

superficial (but also more famous) treatment of the natural
numbers, employing the new symbolic language of logic,
and Gottlob Frege elaborated his own ideas, which however
fell prey to the paradoxes. An important book inspired by
the set-theoretic style of thinking was Hilbert’s Grundlagen
der Geometrie (1899), which took the “mathematics of
axioms” one step beyond Dedekind through a rich study of
geometric systems motivated by questions concerning the
independence of his axioms. Hilbert’s book made clear the
new axiomatic methodology that had been shaping up in
connection with the novel methods of set theory, and he
combined it with the axiomatic trends coming from
projective geometry.

Nevertheless, as we said before, there was quite a lot of

criticism of set-theoretic, infinitarian methods. As early as
1870, Kronecker had begun to voice critical remarks of a
constructivist bent that, many years later, would be echoed
by prominent thinkers like Brouwer or Wittgenstein.
Kronecker’s critical orientation pointed in the way of
renouncing the real number system and classical analysis, in
favor of some more stringent form of analysis — twentieth
century examples of this would be predicative analysis (H.
Weyl building on basic notions of Poincaré, see Feferman
1988) and intuitionistic analysis (Brouwer). Even Weierstrass
had objections (in 1874, at least) against the idea of
distinguishing sizes of infinity, and that on the face of
Cantor’s proofs. Examples abound, and so during the 1900s
many mathematicians expressed doubts about key ideas and
methods of set theory. A prototype case is E. Borel, who after
introducing the ideas of Cantor in France [1898], became
increasingly suspicious of set theory (the five letters
exchanged by him and Baire, Lebesgue, Hadamard in 1905
have become famous; see Ewald [1996, vol. 2]). But there are
also the cases of Poincaré, Weyl, Skolem, and so on. Among
philosophers, the most prominent example is Wittgenstein,
who condemned set theory for building on the "nonsense"
of fictitious symbolism, suggesting “wrong imagery”, and
so on.

3. Critical Period
In the late nineteenth century, it was a widespread idea that
pure mathematics is nothing but an elaborate form of
arithmetic. Thus it was usual to talk about the
“arithmetisation” of mathematics, and how it had brought
about the highest standards of rigor. With Dedekind and
Hilbert, this viewpoint led to the idea of grounding all of
pure mathematics in set theory. The most difficult steps in
bringing forth this viewpoint had been the establishment of
a theory of the real numbers, and a set-theoretic reduction
of the natural numbers. Both problems had been solved by
the work of Cantor and Dedekind. But precisely when
mathematicians were celebrating that “full rigor” had been
finally attained, serious problems emerged for the
foundations of set theory. First Cantor, and then Russell,
discovered the paradoxes in set theory.

Cantor was led to the paradoxes by having introduced the

“conceptual sphere” of the transfinite numbers. Each
transfinite ordinal is the order type of the set of its
predecessors; e.g., ω is the order type of {0, 1, 2, 3, …}, and
ω + 2 is the order type of {0, 1, 2, 3, … , ω, ω + 1}. Thus, to each

initial segment of the series of ordinals, there corresponds an

immediately greater ordinal. Now, the “whole series” of all
transfinite ordinals would form a well-ordered set, and to it
there would correspond a new ordinal number. This is
unacceptable, for this ordinal o would have to be greater
than all members of the “whole series”, and in particular
o < o. This is usually called the Burali-Forti paradox, or

paradox of the ordinals (although Burali-Forti himself failed

to formulate it clearly, see Moore & Garciadiego 1981).

Although it is conceivable that Cantor might have found that

paradox as early as 1883, immediately after introducing the
transfinite ordinals (for arguments in favour of this idea see
Purkert & Ilgauds 1987 and Tait 2000), the evidence indicates
clearly that it was not until 1896/97 that he found this
paradoxical argument and realized its implications. By this
time, he was also able to employ Cantor’s Theorem to yield
the Cantor paradox, or paradox of the alephs: if there existed
a “set of all” cardinal numbers (alephs), Cantor’s Theorem
applied to it would give a new aleph ℵ, such that ℵ < ℵ. The
great set theorist realized perfectly well that these paradoxes
were a fatal blow to the “logical” approaches to sets
favoured by Frege and Dedekind. Cantor emphasized that
his views were “in diametrical opposition” to Dedekind’s,
and in particular to his “naïve assumption that all well-
defined collections, or systems, are also ‘consistent
systems’ ” (see the letter to Hilbert, Nov. 15, 1899, in
Purkert & Ilgauds 1987: 154). (Contrary to what has often
been claimed, Cantor’s ambiguous definition of set in his
paper of 1895 was intended to be “diametrically opposite”
to the logicists’ understanding of sets—often called
“naïve” set theory, which could more properly be called
the dichotomy conception of sets, following a suggestion of
Gödel.)

Cantor thought he could solve the problem of the paradoxes

by distinguishing between “consistent multiplicities” or
sets, and “inconsistent multiplicities”. But, in the absence
of explicit criteria for the distinction, this was simply a verbal
answer to the problem. Being aware of deficiencies in his
new ideas, Cantor never published a last paper he had been
preparing, in which he planned to discuss the paradoxes and
the problem of well-ordering (we know quite well the
contents of this unpublished paper, as Cantor discussed it in
correspondence with Dedekind and Hilbert; see the 1899
letters to Dedekind in Cantor 1932, or Ewald 1996: vol. 2).
Cantor presented an argument that relied on the “Burali-
Forti” paradox of the ordinals, and aimed to prove that
every set can be well-ordered. This argument was later
rediscovered by the British mathematician P.E.B. Jourdain,
but it is open to criticism because it works with
“inconsistent multiplicities” (Cantor’s term in the above-
mentioned letters).
Cantor’s paradoxes convinced Hilbert and Dedekind that
there were important doubts concerning the foundations of
set theory. Hilbert formulated a paradox of his own
(Peckhaus & Kahle 2002), and discussed the problem with
mathematicians in his Göttingen circle. Ernst Zermelo was
thus led to discover the paradox of the “set” of all sets that
are not members of themselves (Rang & Thomas 1981). This
was independently discovered by Bertrand Russell, who was
led to it by a careful study of Cantor’s Theorem, which
conflicted deeply with Russell’s belief in a universal set.
Some time later, in June 1902, he communicated the
“contradiction” to Gottlob Frege, who was completing his
own logical foundation of arithmetic, in a well-known letter
[van Heijenoort 1967, 124]. Frege’s reaction made very
clear the profound impact of this contradiction upon the
logicist program. “Can I always speak of a class, of the
extension of a concept? And if not, how can I know the
exceptions?” Faced with this, “I cannot see how arithmetic
could be given a scientific foundation, how numbers could
be conceived as logical objects” (Frege 1903: 253).

The publication of Volume II of Frege’s Grundgesetze

(1903), and above all Russell’s work The Principles of
Mathematics (1903), made the mathematical community
fully aware of the existence of the set-theoretic paradoxes, of
their impact and importance. There is evidence that, up to
then, even Hilbert and Zermelo had not fully appreciated the
damage. Notice that the Russell-Zermelo paradox operates
with very basic notions—negation and set membership—
concepts that had widely been regarded as purely logical.
The “set” R = {x : x ∉ x} exists according to the principle
of comprehension (which allows any open sentence to
determine a class), but if so, R ∈ R iff R ∉ R. It is a direct
contradiction to the principle favoured by Frege and Russell.

It was obviously necessary to clarify the foundations of set

theory, but the overall situation did not make this an easy
task. The different competing viewpoints were widely
divergent. Cantor had a metaphysical understanding of set
theory and, although he had one of the sharpest views of the
field, he could not offer a precise foundation. It was clear to
him (as it had been, somewhat mysteriously, to Ernst
Schröder in his Vorlesungen über die Algebra der Logik,
1891) that one has to reject the idea of a Universal Set,
favoured by Frege and Dedekind. Frege and Russell based
their approach on the principle of comprehension, which was
shown contradictory. Dedekind avoided that principle, but
he postulated that the Absolute Universe was a set, a
“thing” in his technical sense of Gedankending; and he
coupled that assumption with full acceptance of arbitrary
subsets.

This idea of admitting arbitrary subsets had been one of the

deep inspirations of both Cantor and Dedekind, but none of
them had thematized it. (Here, their modern understanding
of analysis played a crucial but implicit background role,
since they worked within the Dirichlet-Riemann tradition of
“arbitrary” functions.) As for the now famous iterative
conception there were some elements of it (particularly in
Dedekind’s work, with his iterative development of the
number system, and his views on “systems” and
“things”), but it was conspicuously absent from many of
the relevant authors. Typically, e. g., Cantor did not iterate
the process of set formation: he tended to consider sets of
homogeneous elements, elements which were taken to
belong “in some conceptual sphere” (either numbers, or
points, or functions, or even physical particles—but not
intermingled). The iterative conception was first suggested
by Kurt Gödel in [1933], in connection with technical work by
von Neumann and Zermelo a few years earlier; Gödel would
insist on the idea in his well-known paper on Cantor’s
continuum problem. It came only post facto, after very
substantial amounts of set theory had been developed and
fully systematized.

This variety of conflicting viewpoints contributed much to

the overall confusion, but there was more. In addition to the
paradoxes discussed above (set-theoretic paradoxes, as we
say), the list of “logical” paradoxes included a whole array
of further ones (later called “semantic”). Among these are
paradoxes due to Russell, Richard, König, Berry, Grelling, etc.,
as well as the ancient liar paradox due to Epimenides. And
the diagnoses and proposed cures for the damage were
tremendously varied. Some authors, like Russell, thought it
was essential to find a new logical system that could solve all
the paradoxes at once. This led him into the ramified type
theory that formed the basis of Principia Mathematica (3
volumes, Whitehead and Russell 1910–1913), his joint work
with Alfred Whitehead. Other authors, like Zermelo, believed
that most of those paradoxes dissolved as soon as one
worked within a restricted axiomatic system. They
concentrated on the “set-theoretic” paradoxes (as we have
done above), and were led to search for axiomatic systems of
set theory.

Even more importantly, the questions left open by Cantor

and emphasized by Hilbert in his first problem of 1900
caused heated debate. At the International Congress of
Mathematicians at Heidelberg, 1904, Gyula (Julius) König
proposed a very detailed proof that the cardinality of the
continuum cannot be any of Cantor’s alephs. His proof was
only flawed because he had relied on a result previously
“proven” by Felix Bernstein, a student of Cantor and
Hilbert. It took some months for Felix Hausdorff to identify
the flaw and correct it by properly stating the special
conditions under which Bernstein’s result was valid (see
Hausdorff 2001, vol. 1). Once thus corrected, König’s
theorem became one of the very few results restricting the
possible solutions of the continuum problem, implying, e.g.,
that card(R) is not equal to ℵ . Meanwhile, Zermelo was able
ω

to present a proof that every set can be well-ordered, using

the Axiom of Choice [1904]. During the following year,
prominent mathematicians in Germany, France, Italy and
England discussed the Axiom of Choice and its acceptability.

The Axiom of Choice states: For every set A of non-empty

sets, there exists a set that has exactly one element in
common with each set in A. This started a whole era during
which the Axiom of Choice was treated most carefully as a
dubious hypothesis (see the monumental study by Moore
1982). And that is ironic, for, among all of the usual
principles of set theory, the Axiom of Choice is the only one
that explicitly enforces the existence of some arbitrary
subsets. But, important as this idea had been in motivating
Cantor and Dedekind, and however entangled it is with
classical analysis, infinite arbitrary subsets were rejected by
many other authors. Among the most influential ones in the
following period, one ought to emphasize the names of
Russell, Hermann Weyl, and of course Brouwer.

Choice was, for a long time, a controversial axiom. On the

one hand, it is of wide use in mathematics and, indeed, it’s
key to many important theorems of analysis (this became
gradually clear with works such as Sierpiński [1918]). On the
other hand, it has rather unintuitive consequences, such as
the Banach-Tarski Paradox, which says that the unit ball can
be partitioned into finitely-many ‘pieces’ (subsets), which
can then be rearranged to form two unit balls (see
Tomkowicz & Wagon [2019]). The objections to the axiom
arise from the fact that it asserts the existence of sets that
cannot be explicitly defined. During the 1920s and 1930s,
there existed the ritual practice of mentioning it explicitly,
whenever a theorem would depend on the axiom. This
stopped only after Gödel’s proof of relative consistency,
discussed below.

The impressive polemics which surrounded his Well-

Ordering Theorem, and the most interesting and difficult
problem posed by the foundations of mathematics, led
Zermelo to concentrate on axiomatic set theory. As a result
of his incisive analysis, in 1908 he published his axiom
system, showing how it blocked the known paradoxes and
yet allowed for a masterful development of the theory of
cardinals and ordinals. This, however, is the topic of the entry
Zermelo’s axiomatization of set theory; also, on the life and
work of Zermelo, see Ebbinghaus 2015.

4. From Zermelo to Gödel

In the period 1900–1930, the rubric “set theory” was still
understood to include topics in topology and the theory of
functions. Although Cantor, Dedekind, and Zermelo had left
that stage behind to concentrate on pure set theory, for
mathematicians at large this would still take a long time.
Thus, at the first International Congress of Mathematicians,
1897, keynote speeches given by Hadamard and Hurwitz
defended set theory on the basis of its importance for
analysis. Around 1900, motivated by topics in analysis,
important work was done by three French experts: Borel
[1898], Baire [1899] and Lebesgue [1902] [1905]. Their work
inaugurated the development of descriptive set theory by
extending Cantor’s studies on definable sets of real
numbers (in which he had established that the Continuum
Hypothesis is valid for closed sets). They introduced the
hierarchy of Borel sets, the Baire hierarchy of functions, and
the concept of Lebesgue measure—a crucial concept of
modern analysis.

Descriptive set theory (DST) is the study of certain kinds of

definable sets of real numbers, which are obtained from
simple kinds (like the open sets and the closed sets) by well-
understood operations like complementation or projection.
The Borel sets were the first hierarchy of definable sets,
introduced in the 1898 book of Émile Borel; they are
obtained from the open sets by iterated application of the
operations of countable union and complementation. In
1905 Lebesgue studied the Borel sets in an epochal memoir,
showing that their hierarchy has levels for all countable
ordinals, and analyzing the Baire functions as counterparts of
the Borel sets. The main aim of descriptive set theory is to
find structural properties common to all such definable sets:
for instance, the Borel sets were shown to have the perfect
set property (if uncountable, they have a perfect subset) and
thus to comply with the continuum hypothesis (CH). This
result was established in 1916 by Hausdorff and by
Alexandroff, working independently. Other important
“regularity properties” studied in DST are the property of
being Lebesgue measurable, and the so-called property of
Baire (to differ from an open set by a so-called meager set,
or set of first category).
Also crucial at the time was the study of the analytic sets,
namely the continuous images of Borel sets, or equivalently,
the projections of Borel sets. The young Russian
mathematician Mikhail Suslin found a mistake in
Lebesgue’s 1905 memoir when he realized that the
projection of a Borel set is not Borel in general [Suslin 1917].
However, he was able to establish that the analytic sets, too,
possess the perfect set property and thus verify CH. By 1923
Nikolai Lusin and Wacław Sierpiński were studying the co-
analytic sets, and this was to lead them to a new hierarchy of
projective sets, which starts with the analytic sets (Σ ), their
1
1

complements (co-analytic, Π sets), the projections of these

1
1

last (Σ sets), their complements (Π sets), and so on. During

1
2
1
2

the 1920s much work was done on these new types of sets,
mainly by Polish mathematicians around Sierpiński and by
the Russian school of Lusin and his students. A crucial result
obtained by Sierpiński was that every Σ set is the union of
1
2

ℵ Borel sets (the same holds for Σ sets), but this kind of
1
1 1

traditional research on the topic would stagnate after around

1940 (see Kanamori [1995]).

Soon Lusin, Sierpiński and their colleagues were finding

extreme difficulties in their work. Lusin was so much in
despair that, in a paper of 1925, he came to the “totally
unexpected” conclusion that “one does not know and one
will never know” whether the projective sets have the
desired regularity properties (quoted in Kanamori 1995: 250).
Such comments are highly interesting in the light of later
developments, which have led to hypotheses that solve all
the relevant questions (Projective Determinacy, in particular).
They underscore the difficult methodological and
philosophical issues raised by these more recent hypotheses,
namely the problem concerning the kind of evidence that
backs them.

Lusin summarized the state of the art in his 1930 book

Leçons sur les ensembles analytiques (Paris, Gauthier-Villars),
which was to be a key reference for years to come. Since this
work, it has become customary to present results in DST for
the Baire space ω of infinite sequences of natural numbers,
ω

which in effect had been introduced by René Baire in a paper

published in 1909. Baire space is endowed with a certain
topology that makes it homeomorphic to the set of the
irrational numbers, and it is regarded by experts to be
“perhaps the most fundamental object of study of set
theory” next to the set of natural numbers [Moschovakis
1994, 135].

This stream of work on DST must be counted among the

most important contributions made by set theory to analysis
and topology. But what had begun as an attempt to prove
the Continuum Hypothesis could not reach this goal. Soon it
was shown using the Axiom of Choice that there are non-
Lebesgue measurable sets of reals (Vitali 1905), and also
uncountable sets of reals with no perfect subset (Bernstein
1908). Such results made clear the impossibility of reaching
the goal of CH by concentrating on definable and “well-
behaved” sets of reals.

Also, with Gödel’s work around 1940 (and also with forcing
in the 1960s) it became clear why the research of the 1920s
and 30s had stagnated: the fundamental new independence
results showed that the theorems established by Suslin
(perfect set property for analytic sets), Sierpinski (Σ sets as
1
2

unions of ℵ Borel sets) and a few others were the best

1

possible results on the basis of axiom system ZFC. This is

important philosophically: already an exploration of the
world of sets definable from the open (or closed) sets by
complement, countable union, and projection had sufficed
to reach the limits of the ZFC system. Hence the need for
new axioms, that Gödel emphasized after World War II
[Gödel 1947].

Let us now turn to Cantor’s other main legacy, the study of

transfinite numbers. By 1908 Hausdorff was working on
uncountable order types and introduced the Generalized
Continuum Hypothesis (2 = ℵ ). He was also the first to
ℵa
a+1

consider the possibility of an “exorbitant” cardinal, namely

a weakly inaccessible, i.e., a regular cardinal that is not a
successor (a cardinal α is called regular if decomposing α
into a sum of smaller cardinals requires α-many such
numbers). Few years later, in the early 1910s, Paul Mahlo was
studying hierarchies of such large cardinals in work that
pioneered what was to become a central area of set theory;
he obtained a succession of inaccessible cardinals by
employing a certain operation that involves the notion of a
stationary subset; they are called Mahlo cardinals. But the
study of large cardinals developed slowly. Meanwhile,
Hausdorff’s textbook Grundzüge der Mengenlehre (1914)
introduced two generations of mathematicians into set
theory and general topology.

The next crucial steps into the “very high” infinite were
done in 1930. The notion of strongly inaccessible cardinals
was then isolated by Sierpiński & Tarski, and by Zermelo
[1930]. A strong inaccessible is a regular cardinal α such that
2 is less than α whenever x < α. While weak inaccessibles
x

merely involve closure under the successor operation, strong

inaccessibles involve a much stronger notion of closure
under the powerset operation. That same year, in a path-
breaking paper on models of ZFC, Zermelo [1930]
established a link between the uncountable (strongly)
inaccessible cardinals and certain “natural” models of ZFC
(in which work he assumed that the powerset operation is,
so to say, fully determinate).

In that same year, Stanislaw Ulam was led by considerations

coming out of analysis (measure theory) to a concept that
was to become central: measurable cardinals. It turned out
that such cardinals, defined by a measure-theoretic property,
had to be (strongly) inaccessible. Indeed, many years later it
would be established (by Hanf, working upon Tarski’s
earlier work) that the first inaccessible cardinal is not
measurable, showing that these new cardinals were even
more “exorbitant”. As one can see, the Polish school led by
Sierpiński had a very central role in the development of set
theory between the Wars. Measurable cardinals came to
special prominence in the late 1960s when it became clear
that the existence of a measurable cardinal contradicts
Gödel’s axiom of constructibility (V = L in the class
notation). This again vindicated Gödel’s convictions,
expressed in what is sometimes called “Gödel’s program”
for new axioms.

Set-theoretic mathematics continued its development into

the powerful axiomatic and structural approach that was to
dominate much of the 20th century. To give just a couple of
examples, Hilbert’s early axiomatic work (e.g., in his arch-
famous Foundations of Geometry) was deeply set-theoretic;
Ernst Steinitz published in 1910 his research on abstract field
theory, making essential use of the Axiom of Choice; and
around the same time the study of function spaces began
with work by Hilbert, Maurice Fréchet, and others. During
the 1920s and 30s, the first specialized mathematics journal,
Fundamenta Mathematicae, was devoted to set theory as
then understood (centrally including topology and function
theory). In those decades structural algebra came of age,
abstract topology was gradually becoming an independent
branch of study, and the study of set theory initiated its
metatheoretic turn.

Ever since, “set theory” has generally been identified with

the branch of mathematical logic that studies transfinite sets,
originating in Cantor’s result that R has a greater
cardinality than N. But, as the foregoing discussion shows,
set theory was both effect and cause of the rise of modern
mathematics: the traces of this origin are indelibly stamped
on its axiomatic structure.

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Further Reading

Cavaillès, Jean, 1962, Philosophie mathématique, Paris:

Hermann.
Ebbinghaus, Heinz-Dieter, 2007, Ernst Zermelo: An approach
to his life an work, New York: Springer.
Fraenkel, Abraham, 1928, Einleitung in die Mengenlehre, 3rd
edn. Berlin: Springer.
Grattan-Guinness, Ivor (ed.), 1980, From the Calculus to Set
Theory, 1630–1910, London: Duckworth.
Kanamori, Akihiro, 2004, “Zermelo and set theory”, Bulletin
of Symbolic Logic, 10(4): 487–553.
–––, 2007, “Gödel and set theory”, Bulletin of Symbolic
Logic, 13 (2): 153–188.
–––, 2008, “Cohen and set theory”, Bulletin of Symbolic
Logic, 14(3): 351–378.
–––, 2009, “Bernays and set theory”, Bulletin of Symbolic
Logic, 15(1): 43–60.
Maddy, Penelope, 1988, “Believing the axioms”, Journal of
Symbolic Logic, 53(2): 481–511; 53(3): 736–764.
Wagon, Stan, 1993, The Banach-Tarski Paradox, Cambridge:
Cambridge University Press.
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Other Internet Resources

A History of Set Theory, by J.J. O’Connor and E.F.
Robertson, in The MacTutor History of Mathematics
archive. Note that their reconstruction conflicts at some
points with the one provided here.
Godel’s Program (PowerPoint), an interesting talk by
John R. Steel (Mathematics, U.C./Berkeley).
A Home Page for the Axiom of Choice, maintained by Eric
Schechter (Mathematics, Vanderbilt University).

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