72 Russell summer 1989 Reviews 73

Introducing Kurt Godel the other authors included in this volume.

Dawson's seven-page G6del chronology is superior to any of the Russell
chronologies I have seen, and should prove useful for Russell scholars who wish
by Billy Joe Lucas to integrate the sequence of events in G6del's life into their image of the sequen-
tial flow of Russell's life and work.
Readers with skill at reading photographs will take pleasure in the six well-
Kurt G6del. Collected Works. Volume I: Publications 1929-1936. Ed. Solomon reproduced photos contained in this volume. These are not listed in the table
Feferman (Ed.-in-chief), John W. Dawson, Jr., Stephen C. Kleene, Gregory of contents; one occurs opposite the title-page, and the others between pages
H. Moore, Robert M. Solovay, and Jean van Heijenoort. Prepared under the 15 and 16.
auspices of the Association for Symbolic Logic. New York: Oxford University The third largest portion of this book, after the 145 pages of G6del in English
Press; Oxford: Clarendon Press, 1986. Pp. xvi, 474. US$35.00. and the 119 pages of G6del in German, is the approximately seventy-eight pages
comprised of nineteen introductory notes to some thirty-five of the fifty-six
works of G6del collected in this volume. These nineteen introductory notes,
easily distinguished from G6del's text by the occurrence of a vertical line down
EVAL UATING RussELL's CONTRIBUTIONS to logic, his applications of the outside margin, were written by the following eleven authors: two are by
logic, and his views on the nature of logic and on the relationship of logic to John W. Dawson, Jr.; one is co-authored by Burton Dreben and Jean van Hei-
philosophy invQlves answering some serious questions concerning twentieth- jenoort; one is by Solomon Feferman; one is by Warren D. Goldfarb; three are
century logic, logic both as Russell knew it and as it has evolved since Russell by Stephen C. Kleene; one is by Rohit Parikh; five are by W.V. Quine; three
made his contributions. As G6del is second only to Russell among contributors are by A.S. Troelstra; one is by Robert L. Vaught; and one is by Judson Webb.
to logic in the first half of this century, our understanding of Russell will thus Solomon Feferman has contributed a thirty-six page essay, "G6del's Life and
be incomplete until we come to appreciate fully the work of Kurt G6del. Work," some fourteen pages of which are an account of G6del's personal life
Understanding G6del is a task that should become easier for most Russell . and career, and the remainder of which contains both a brief description of the
students now that G6del's collected works are being published in English trans- contents of this and subsequent volumes of G6del's works, as well as an acces-
lation. Volume I contains, or rather was intended to contain (see Section VI sible overview not only of G6del's contributions to logic but also of his work
below), all of G6del's writings published in the period 1929-1936. Volume II and thought in general. This reviewer, as might be expected, ~annot agree with
is to contain the remainder of G6del's published works. Subsequent volumes Feferman's reference (on p. I) to G6del as "the most important logician of the
are to contain his "unpublished manuscripts, lectures, lecture notes, and cor- twentieth century",· but making the case for Russell~s right to be the referent
respondence, as well as extracts from his scientific notebooks." of this description is not, of course, within the proper scope of this review essay.
Except for his 1934 lectures on undecidability (which were published orig- Russell scholars who read Feferman's essay will no doubt be struck by the
inally in English), all of G6del's publications as of 1936 were in German. following parallel between the lives of G6del and Russell: nearly all of the work
G6del's German texts are printed on left-hand pages, with English translations in logic for which either is famous was done in an eleven-year period-190o-
on the facing pages. Except for his dissertation, original pagination is retained 1910 for Russell, and 1929-1939 for G6del. (The parallel continues: both did
in margin notes, and occasional footnotes (mostly G6del's) are located at the somewhat less well-known work later.)
bottom of the page. G6del also wrote, as did Russell, many reviews of technical books early in
There are 490 numbered pages in this book. In English, the works of G6del his career. Of the fifty-six works of G6del collected in Volume I, thirty-three
collected in this volume total approximately 145 pages, or thirty percent of the are reviews published from 1931 to 1936. Of these, nineteen are of less than
total. G6del's works, in German, come to another 119 pages; G6del's writings half a page in length (five occupy less than ten lines of print each), and only
thus comprise about fifty-four percent of the printed material. The section one requires more than one page of print: G6del's two-page-plus 1933 review
titled "Textual Notes" occupies only five pages, and fails to mention that of the revised edition of his former teacher Hans Hahn's book Reelle Funkti-
G6del's citations of publications have been altered, sometimes both in German onen. 1 Six of these reviews appeared in the Monatshefte fur Mathematik und Phy-
originals and in the translations thereof, to conform to the style used by other sik, four of them in 1931. The remaining twenty-seven were all published in
contributors to this volume. This is mentioned in a four-page section. titled, the Zentralblatt fur Mathematik und ihre Grenzgebiete, twenty-four of them in
"Information for the Reader:" The only index is a fourteen-page index of the years 1932-35. Another parallel between the careers of G6del and Russell:
names of persons, places, and publications that fails to differentiate between
occurrences of names in G6del's writings and occurrences in the writings of
1 Leipzig: 1932.
74 . Russell summer 1989 Reviews 75

although both were experienced reviewers, neither ever wrote reviews for van Heijenoort (1931 and 1932), are reproduced here, along with Godel's "On
Mathematical Reviews or The Journal of Symbolic Logic. Dawson2 notes that Undecidable Propositions of Formal Mathematical Systems". The latter con-
Godel was not even asked to review for Mathematical Reviews until 1962. It sists of notes taken in 1934, by Kleene and J. Barkley Rosser, of Godel's lec-
would be interesting to know whether Russell was ever asked to review for tures at the Institute for Advanced Study, together with Godel's later postscript
either journal. and corrections. All four of these works have been readily available in English
II for some time, although not all together in one volume. Kleene's fifteen-page
introduction to the 1930, 1931 and 1932 papers on incompleteness is a model
The first three works of Godel in this volume are his dissertation of 1929 introduction: it sets Godel's work in proper historical context, prepares the
(twenty-one pages in English), a revised and substantially abbreviated version reader to understand Godel's text, and relates Godel's papers to important
(eleven pages in English) published in 1930, and a brief abstract based on a developments that have followed. Kleene's eight-page introduction to the 1934
presentation of Godel's results in Konigsberg on 6 September 1930. Of all of lecture notes is by itself a sufficient reason to purchase this volume.
Godel's longer, published writings, his dissertation has been, until now, the
most difficult to obtain, and is here translated for the first time into English, IV
by Stefan Bauer-Mengelberg and van Heijenoort. The 1930 essay is translated
by Bauer-Mengelberg, with suggestions and revisions by Godel and van Hei- In addition to these two groups of essays (three on completeness, and four on
jenoort. The 1930 abstract is translated by pawson. Dreben and van Heijenoort incompleteness), Volume I contains forfy-nine additional items written by
have contributed a sixteen-page introduction to these three works in which they Godel, twenty-seven of them with accompanying introductions. These other
interpret the historical context in which Godel saw completeness as a problem writings of Godel vary both in topic and in length.
and suggest just what his philosophical goals were, sort out some of the tangled The topics covered are diverse and include: Skolem's work on non-standard
history of the origins of the completeness, compactness, and (downward) Low- models for arithmetic (Vaught has contributed a three-page, compact analysis
enheim-Skolem theorems (all of which are proven either in Godel's dissertation of both Skolem and Godel's reviews of Skolem); Hilbert's introduction of a
or in the 1930 article), and describe some connections between Godel's work restricted, informal "rule" of infinite induction for use in proving completeness
on completeness and more recent developments in logic. Godel's way of prov- (Feferman's five-page introduction to Godel's review of Hilbert discusses both
ing completeness is easily misunderstood: logicians of the calibre ,of Hilbert, the meaning of Hilbert's new rule and the "personal" side of Godel's relation-
Ackermann, and Church have erred in expounding it. In addition, then, to ship to Hilbert); Church's early work on foundational lambda calculus that was
Godel's two versions of the proof and Dreben and van Heijenoort's introduc- intended as an improvement over Russell's type theory (Godel's three reviews
tion in this volume, the beginning reader may wish to consult section 44 of of Church are introduced by Kleene); and Godel's statement, without proof,
Church. 3 Of special interest to Russell scholars is the fact that Godel's results of his 1936 "theorem" concerning the practical merits of type theory (Parikh's
are based on the first-order fragment of Principia Mathematica, with axiom *1.5 two-page introduction to the topic of Godel's speed-up theorem is a master-
deleted and axiom *1.3 replaced by theorem *2.2, and are, therefore, among piece). In addition, Godel published five short, highly technical papers on
the earliest results describing aspects of Russell and Whitehead's magnum opus. geometry (average length: one page) in Ergebnisse eines mathematischen Kollo-
Also worth noting is the fact that Godel here proves the independence of each quiums in 1933. Webb has written an interesting four-page technical introduc-
of the first-order axioms and rules of Principia, except, of course, for *1.5, tion that puts these notes into an historical context, while relating them both
which Bernays had already shown (in 1926) to be derivable from the others. to later developments in geometry and to Godel's views on completeness. Judg-
ing only from his published work, the closest Godel ever came to returning to
III work in geometry was in his papers of 1949 and 1952 on physics.
Excluding publications mentioned in earlier sections, Godel's writings vary
Due in part to widespread false claims about how Godel's incompleteness theo- in length from that of his eleven-page 1933 essay on the decision problem for
rem refutes Russell's logistic thesis, perhaps the writings of Godel best known first-order logic (which is accompanied by Goldfarb's elegant and helpful five-
to those who study Russell are his three papers on incompleteness 'published page introduction to this and two other publications on the decision problem),
in 1930, 1931, and 1932. These, as translated by Bauer-Mengelberg (1930) and to that of the following untitled remark published in Ergebnisse eines mathe-
matischen Kolloquiums in 1936:
2 John W. Dawson, Jr., "The Published Work of Kurt GOdel: an Annotated Bibliography," Notre
Dame Journal of Formal Logic, 24 (1983): 255-84 (at 272). Actually, for each individual entrepreneur the demand also depends on his income,
3 Alonzo Church, Introduction to Mathematical Logic, Vol. I (Princeton: Princeton University Press,
and that in turn depends on the price of the factors of production. One can formulate
195 6).
76 Russell summer 1989 Reviews 77

an appropriate system of equations and investigate whether it is solvable. into theorems provable by intuitionistic methods. Troelstra's four-page intro-
duction to this essay explains G6del's method of translation (there is what can
Dawson, in his one-page introduction' to this remark, not only manages to set only be a typographical error in Troelstra's presentation-see the list of errata
it in a context in which it makes sense, but also provides the reader with ref- at the end of this review), and sets G6del's work in the context both of earlier
erences that substantiate the claim that G6del had a not insignificant influence anticipations and of subsequent extensions of his results. Students of Principia
on Oskar Morgenstern's work in economics. (Serious Russell students will may find Troelstra's remarks concerning the proof of the consistency of G6del's
recall, of course, Russell's relationship with Keynes.) . system of classical arithmetic relative to intuitionistic mathematics of special
Although G6del wrote nowhere near as much as Russell on geometry, phys- interest. G6del's other 1933 paper on intuitionism is discussed in the following
ics, or economics, this collection of GOOel's papers makesjit possible for us to section on modal logic.
see that, like Russell, G6del's interest in mathematics was not restricted to work
that verified or refuted various philosophical claims, but also included an inter- VI
est both in math for math's sake and in substantive uses of mathematics to
increase our understanding of the world in which we live. At least four works of G6del included in this volume are concerned with modal
logic: his brief 1931 review of Oskar Becker's "On the Logic of Modalities"
v (this is a suggested correction to the translation of Becker's title in this volume);
his article, "An Interpretation of the Intuitionistic Propositional Calculus"
Neither Russell nor G6del accepted the intuitionist philosophy of mathematics (1933); his 1933 review of Lewis;5 and his 1935 review of Huntington. 6 In terms
associated with Brouwer and Heyting, yet both showed a long-term preoccu- of influence on subsequent developments within a subfield of logic, G6del's
pation with it. The opening section of G6del's first work (his dissertation of work on modal logic is not as significant as his work on recursive functions and
1929) reveals his concern with intuitionism,4 and this concern is paramount in on set theory, yet it deserves more extensive study than it receives here. Two
four of the works published in this volume, two of which were first published of these works (the review of Becker and the review of Lewis) are not discussed
in 1932, and two in 1933. in any of the twenty introductions to various aspects of G6del's thought. Two
Of G6del's two 1932 publications on intuitionism, his "On the Intuitionistic are.
Propositional Calculus" is far and away the most impressive. Troelstra's intro- Troelstra has contributed a fine three-page introduction to G6del's one-page
duction to this stunning one-page paper realizes the following goals. In no more note of 1933 in which G6del shows how to translate effectively any formula A
than one page the reader is, among other things, given the necessary back- of Heyting's intuitionistic logic into a formula A' of G6del's modal logic S4 in
ground to understand this rich little paper by G6del, told how G6del's results such a way that A is a theorem of Heyting's system if and only if A I is a theorem
relate to later work on intermediate logics by Ivo Thomas and Michael Dum- of S4. Troelstra's essay, while describing both G6del's achievements and sub-
mett, informed as to which result in G6del's essay constitutes the first result sequent related developments in the field of provability modal logics, manages
on intermediate propositional logics, given some insight into the ways in which to unobtrusively integrate references to some seventeen important publications
tht:se la.tter logics are, and are not, significant, and referred to excellent recent that either influenced or anticipated G6del or continued the line of work begun
technical surveys and annotated bibliographies on intermediate (between intui- in his 1933 essay.
tionistic and classical) logics and on many-valued logics. Except for Troelstra's elegant introduction to G6del's note on the interpre-
G6del's five-page 1933 paper, "On Intuitionistic Arithmetic and Number tation of intuitionistic propositional calculus, and Feferman's discussion of it
Theory", shows how to translate every formula of a system of classical arith- in his general introduction, G6del's contributions to modal logic are not placed
metic into a formula of intuitionistic mathematics in such a way as to be able in historical context, nor is the reader given any reference to subsequent work
to prove that all. the theorems provable in the classical system are translated in the field. The only other introduction to G6del's publications in the field of
modal logic included in this volume is Quine's introduction to G6del's review
4 Concern with intuitionism is also prominent in the last of Godel's major works to b.e published in of Huntington's "Independent Postulates Related to c.l. Lewis's Theory of
his lifetime, his 1958 essay, "On a Hitherto Unexploited Extension of the Finitary Standpoint". Strict Implication". Quine's introduction consists of these three sentences.
This essay is translated into English by Wilfred Hodges and Bruce Watson in the Journal of Phil-
osophical Logic, 9 (1980): 133-42. Hodges and Watson's translation is accompanied by a bibliog- Huntington's system is not a modal logic, for he uses predicates rather than iterable
raphy (compiled by l.R. Hindley) of work resulting from Godel's paper, and is not included in the
bibliography for this volume. The duration of GOdel's concern with intuitionism is particularly
evident in connection with the history of this essay. Dawson (in "Kurt Godel in Sharper Focus", 5 C.1. Lewis, "Alternative Systems of Logic", The Monist, 42 (1932): 481-5°7.
The Mathematicallmelligencer, 6 [1984]: 16) notes that this paper was based on results Godel had 6 Edward V. Huntington, "Independent Postulates Related to C.l. Lewis's Theory of Strict Impli-
by 1941, and that he worked on an English translation of it as late as the early 1970'S. cation", Mind, n.s. 43 (1934): 181--98.
78 Russell summer 1989 Reviews 79

functors to express necessity and impossibility. Like Lewis, he stops short of quan- VII

tification. But, if he were to introduce it, he would still be unable, on this approach,
to quantify into modal contexts and thus precipitate the referential opacity and related Naturally following upon the question of the completeness of this volume of
perplexities that beset modal logic. Godel's publications is the question of its soundness. An examination both of
the content and of the evidence concerning the authorship of a brief, two-par-
Quine's claim that both Huntington, in his study of Lewis, and Lewis, in agraph note that the editors have included and given the title, "On Parry's
the system studied by Huntington, stop short of quantification is rather per- Axioms", reveals that the second paragraph of this publication is not, in an
plexing. Huntington's postulate 0 (p. 183) and postulate 12 (p. 184) are exis- acceptable sense, the work of Kurt Godel. ll
tential, and hence appear not to stop short of quantification. Furthermore, on
p. 182 of his essay, Huntington says that the axioms of Lewis he is studying VIII
include proposition "(20.01) on page 179" of Lewis and LangfordJ Yet, contra
Quine, Lewis's proposition 20.01 is an existential quantification from outside The bibliography, or list of references, for this volume is decidedly odd. The
the scope of a modal operator of variables that occur within the scope of the following features deserve attention.
modal operator, i.e., Lewis has quantification "into modal contexts". The list of references takes up 53 pages, or just over a third as much space
In addition to the four papers on modal logic included in this volume, Godel as is occupied by Godel's writings in English. Yet, it fails to include such basic
also wrote, in collaboration with William Tuthill Parry, another paper on modal items as the following: a complete list of translations of GOdel's works into
logic that was not included in this volume. On 18 February 1932 Parry gave a English (e.g., neither Meltzer's 1962 translation nor Mendelson's 1965 trans-
lecture at Karl Menger's colloquium in Vienna. Based on Parry's lecture, and lation of Godel's major undecidability paper of 1931 is listed); reviews of earlier
without the use of any notes written by Parry, Godel actually wrote the text of translations of Godel (e.g., Bauer-Mengelberg's 1965 review of Meltzer's trans-
the published report of Parry's lecture. What's more, Godel also contributed lation is not listed); translations of Godel's publications into languages other
original, substantive additions to the material contained in Parry's lecture. The than English (e.g., not even Mosterin,12 the most comprehensive edition of
result of this joint effort, "Zum Lewisschen Aussagenkalkiil", was published Godel to appear in any language as of the date of this volume, is cited); pre-
in Ergebnisse eines mathematischen Kolloquiums in 1933. Godel added a footnote viously published correspondence of Godel (none of the correspondence in
to the report, which said, "the proof given below differs in some respects from Dawson 13 is counted among Godel's publications); some major philosophical
that given by Parry."8 This footnote and my correspondence with Professor and/or expository works on Godel are omitted (e.g., neither Myhill 195214 nor
Parry, from which the account just given is derived and in which Parry says, Nagel and Newman 195815 is here); some landmark papers in the history of
"Godel had, of course, greatly improved my proof; so the paper as published logic that are of major significance in interpreting Godel's work (e.g., Henkin
was a joint paper, and in my (not unbiased) opinion, belongs in the collected 195016 is not included); certain basic texts that would be important for, but
works of Godel",9 seem sufficient to establish "Zum Lewisschen Aussagen- possibly unknown to, readers outside the field of logic, at whom this collection
kalkiil" as a paper co-authored by Godel and Parry. is also aimed (e.g., although Corcoran's revised edition of Tarski's papers is
This collection is, then, incomplete. Godel's hitherto generally unknown col- cited, the inexpensive, paperback edition published by Hackett is not men-
laboration with Parry raises an interesting question: are there other papers in tioned). All of this is compounded by the fact that there are some 316 items
the proceedings of Menger's colloquium during the years Godel served on the entered in the list of references that are not by Godel, and yet are not cited
editorial staff that are also the result of the creative additions of Godel?1O If anywhere in this volume, either by Godel or by the other contributors.
there. are, and if there is no decisive textual or archival evidence. of this in cases There is a questionable choice in the citation of works by Russell: the 1920
where the lecturer in question is now dead, then there is no effective and prac-
tical way to compile a complete edition of Godel's papers.

11 For additional details, see my "Kurt Gooel's Contributions to Relevance Logic", abstract in The
7 Symbolic Logic (New York: Century, 1932). Journal of Symbolic Logic, 54 (1989): 244-5·
12 Ed. Jesus Mosterin, Obras completas (Madrid: 1981).
8 William Tuthill Parry and Kurt GOOel, "Zum Lewisschen Aussagenkalkiil", Ergebnisse eines math-
13 "The Published Work of Kurt Gooel: an Annotated Bibliography".
ematischen Kolloquiums, 4 (1933): 15-16 (at 15)·
9 Letter to Billy Joe Lucas, 31 July 1986. 14 John Myhill, "Some Philosophical Implications of Mathematical Logic", The Review of Meta-
10 On p. 38 of this volume, GOdel is said to have assisted in the editing of volumes 1-5, 7 and 8. In
physics, 6 (1952): 165-98.
Dawson's "Kurt Godel in Sharper Focus", the list includes only volumes 2-5, 7 and 8 (p. 12). 15 Ernest Nagel and James R. Newman, Godel's Proof (New York: New York University Press, 1958).
Gooel's work as reporter and editor deserves further investigation, for the reason just given, as well 16 Leon Henkin, "Completeness in the Theory of Types", The Journal of Symbolic Logic, 15 (1950):
as in connection with the issue raised by section VII below. 81-91.
 80 Russell summer 1989

printing of Introduction to Mathematical Philosophy is incorrectly listed as a "sec-

ond edition" .17
This volume is remarkably free of typographical errors. I have noticed only
the following: p. 183, line 13 for "XI" read "X;"; p. 282, line 15, insert
"( JF)' : = IF"'; p. 354, line 12, delete "<"; p. 386, line 3 of title, p. 387,
line 3 of title, and p. 430 in Huntington 1934 for "Lewis'" read "Lewis's"; p.
442 for Parry 1933a change "17" to "16"; p. 470 to entries under "Parry, Wil-
liam T." add "266".
The production of this volume involved not only the work of the six edltors
and seven additional contributors of introductory notes, but 'also the efforts of,
and support by, a variety of other individuals and organizations. The result is
a well-made, well-organized, landmark book in the history of western intellec-
tual thought. No one's personal library, however small, should be without this
book.

Department of Philosophy
Manhattanville College

17 See Kenneth Blackwell, "A Non-Existent Revision of 'Introduction to Mathematical Philosophy',"

Russell, no. 20 (1975): 16-18.