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~ KURT GODEL April 28, 1906-January 14, 1978 BY STEPHEN C. KLEENE1 TW O P A P E R S ~ ~ 930a, ~ 93 ~ a), both written before the au- thor reached the age of twenty-five, established Kurt Go- de! as second to none among logicians of the modern era, beginning with Frege (18791.2 A third fundamental contri- bution followed a little later ~ ~ 93Xa, ~ 93Sb, ~ 939a, ~ 939b). ORIGINS AND EDUCATION, 1906-1930 Gocle} was born at Brunn in the Austro-Hungarian prov- ince of Moravia. After World War I, Brunn became Brno in Czechoslovakia. Goclel's father Rudolf was managing director and partial owner of one of the leaching textile firms in Brunn; his family had come from Vienna. His mother Mar- ianne had a broad literary education. Her father, Gustav Handschuh, had come from the Rhineland. Godel's family cultivated its German national heritage. After completing secondary school at Brno, in 1924 Go- de] went to Vienna to study physics at the university. The elegant lectures of P. Furtwangler (a pupil of Hilbert and cousin of the famous conductor) fed his interest in mathe ~ For some details of Godel's life, I have drawn upon Kreisel (1980) and Wang (1978, 1981); the authors kindly supplied me with copies. 2 A date shown in parentheses refers to a work listed in the References (or for Godel, in the Bibliography), under the name of the adjacent author. 135

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136 BIOGRAPHICAL MEMOIRS matics, which became his major area of study in 1926. His principal teacher was the analyst Hans Hahn, who was ac- tively interested in the foundations of mathematics. Hahn was a member of the Vienna Circle (Wiener Kreis), the hand of positivist philosophers led by M. Schlick, who was assassi- nated during a lecture in ~ 936. Gocle} attended many of their meetings, without subscribing to all of their doctrines (see Wang 1981, 6531. His doctoral dissertation was completect in the autumn of 1929, and he receiver! the Ph.D. on February 6, 1930. A somewhat reviser! version was presented at Karl Menger's colloquium on May 14, 1930, and was published (1930a) using "several valuable suggestions" of Professor Hahn "regarding the formulation for the publication" (Wang 1981, 654; 1930b is an abstract). GODEL S COMPLETENESS THEOREM ~1930a) Since Frege, the tractitional subject-preclicate analysis of the structure of sentences has been replaced by the more flexible use of propositional functions, or, more briefly, predi- cates.3 Using a given collection or domain D of objects (we call them individuals if they are the primary objects not being analyzecI) as the range of the independent variables, a one- place predicate P or P(a) over D (also callecI a property of mem- bers of D) is a function that, for each member of D as value of the variable a, takes as its value a proposition P(a). A two- place predicate Q or Q(a,b) (also called a binary relation be- tween members of D), for each pair of values of a ant! b from D, takes as value a proposition Q(a,b); ant! so on. In the most commonly cultivated version of logic (the classical logic), the 3 I am endeavoring to give enough background material to enable a scientist who is not a professional mathematician and not already acquainted with mathematical logic to understand Godel's best-known contributions. The memoir (1980) by Krei- sel, about three times the length of the present one, includes many interesting details addressed to mathematicians, if not just to mathematical logicians. The memoir (1978) by Quine gives an excellent overview in just under four pages.

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.. KURT GODEL 137 propositions taken as values of the predicates are each either true or false. The restricted or f rst-order predicate calculus ("elementary logic") deals with expressions, callecIformulas, constructed, in accordance with states! syntactical rules, from: variables a, b, c, ..., x, y, z for individuals; symbols P. Q. R. S. ... for precli- cates; the propositional connectives , ("not"), & ("ancl"), V ("or") and ~ ("implies"~; and the quantifiers Nix ("for all x") and 3x ("Ethere] exists fan] x Esuch thatch. For example, taking P. Q. R to be symbols for predicates of one, two, and three variables, respectively, the expressions P(b), Q(a,c), R(b,a,a), NlxP(x), VxSyQfx,y), Vx(( ,P(x)) ~ Q(a,x)), and Vx(~3yQfy,x)) ~ R(x,a,x)) are formulas. In the classical logic, after making any choice of a non- empty domain D as the range of the variables, each formula can be evaluated as either true or false for each assignment in D of a predicate over D as the value or interpretation of each of its predicate symbols, and of a member of D as the value of each of its "free" variables. Its free variables are the ones with "free" occurrences, where they are not operated upon by quantifiers. In the seven examples of formulas given above, the eight occurrences of a, b, and c are free; the fifteen occurrences of x and y are not free, that is, they are bound. The evaluation process is straightforwarcl, taking V to be the inclusive "or" (A V B is true when one or both of A and B are true, and false otherwise), and handling A ~ B like ~ ~A) V B. For example, taking D to be the non-negative in- tegers or natural numbers i,0, i, 2, ...l, and assigning to P(a), Q(a,b) and R(a,b,c) the predicates "a is even", "a is less than b" and "ab = c", anti to a, b, and c the numbers 0, I, and 1, as values, our seven examples of formulas are respectively false, true, true, false, true, true, and true. Logic is concerned with exploring what formulas express logical truths, that is, are "true in general". LeiLnitz spoke of

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38 BIOGRAPHICAL MEMOIRS truth in all possible worlds. We call a formula valid in D (a given non-empty domain) if it is true for every assignment in D; and simply valid if it is valid in every non-empty do main D. To make reasoning with the predicate calculus practical, paralleling the way we actually think, we cannot stop to think through the evaluation process in all non-empty domains for all assignments each time we want to assure ourselves that a formula is logically true (valid). Instead we use the "axio- matic-deductive method", whereby certain formulas become "provable". First, certain formulas are recognized as being logical ax- ioms. For example, all formulas of either of the following two forms-the forms being called axiom schemata are axioms: A > (B~A). b~xA(x) ~ A(a). Here A, B. and A(x) can be any formulas, and x and a any variables; A(a) is the result of substituting the variable a for the free occurrences of the variable x in the formula A(x). Furthermore, it is required that the resulting occurrences of a in A(a) be free; thus b~x3bQfb,x) > 3bQfb,a) is an axiom by the schema NlxA(x) ~ A(a), but NlxBaQ(a,x) ~ 3aQ(a,a) is not. The axiom schemata (and particular axioms, if we have some not given by schemata) are chosen so that each axiom is valid. Second, circumstances are recognized, called rules of infer- ence, in which, from the one or two formulas shown above the line called premises, the formula shown below the line called the conclusion can be inferred; for example: A, A ~ B B. (I ~ A(x) C ~ 3xA(x). Here A, B. and A(x) can be any formulas, x any variable, and C any formula not containing a free occurrence of x. The

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as KURT GODEL 139 rules of inference are chosen so that, whenever for a given non-empty domain D and assignment in D the premises are true, then for the same D ant! assignment the conclusion is true. Hence, if the premises are valict, the conclusion is valid. A proof is a finite list of formulas, each one in turn being either an axiom or the conclusion of an inference from one or two formulas earlier in the list as the promisers). A proof is a proof of its last formula, which is said to be provable. In one of the stanciard treatments of the classical first- order predicate calculus (Kleene ~ 952, X2), twelve axiom schemata and three rules of inference are usecI.4 By what we have just said about how the axiom schemata (or particular axioms) are chosen (so each axiom is valid), and likewise the rules of inference (so truth is carried for ward by each inference), every provable formula is valid. Thus the axiomatic-deductive treatment of the predicate calculus is correct. But is it complete? That is: Is every vali~formula provable? The axiomatic-clecluctive treatment of the first-orcler predicate calculus, separated out from more complicated log- ical systems, was perhaps first formulated explicitly in Hilbert anct Ackermann's book (19284. The completeness problem was first stated there (p. 681: "Whether the system of axioms [and rules of inference] is complete, so that actually all the logical formulas which are correct for each domain of indi- viduals can be derives! from them, is still an unsolved ques- tion." It is this question that Gode! answered in (1930a). He es- tablished: For each formula A of thefirst-order predicate calculus, either A is provable in it, or A is not valid in the domain 10, I, 2, ...) of the natural numbers (and therefore is not valicl). So, if A is valid, then Goclel's second alternative is excluded, 4 I am giving the version of the predicate calculus with predicate symbols instead of predicate variables, after von Neumann (1927). This I consider easier to explain.

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140 BIOGRAPHICAL MEMOIRS and A is provable. This answers Hilbert and Ackermann's question affirmatively. Let us say that a formula A is (or several formulas are simultaneously) satisf able in a given domain D if A is satisfied (all the formulas are satisfied), that is, made true, by some assignment in D. Then A-is-satisfiable-in-D is equivalent to , A-is-not-valicl-in-D. Now if A is satisf able in some domain D, then ,A is not valid in that domain D, so MA is not valict, so ,A is not provable (by the correctness of the predicate calculus), so by Godel's result appliers to A, MA is not valid in {0, i, 2, ...}, so A is satisfiable in {O. I, 2, ...~. This is a theorem of Lowenheim (1915~. Restating the completeness theorem for ,A: Either MA Is not valid (that is, A is satisfiable) in 10, I, 2, ...), or ,A is provable (equivalently, a contradiction can be cleducecT from A). Gocle} also treated the case for an infinite collection of formulas {Ao' A1, A2, ...} in place of one formula. The result is just what comes from substituting {Ao' A1, A2, ...} for A in the immediately preceding statement, and noting that, if a contradiction can be deducecI from the formulas Ao, A1, A2, ..., only a finite number of them can participate in a given deduction of the contradiction. Thus: Either the formulas Ao, A1. Air ... are simultaneousiv satisfiable in 10, I, 2....T, or, for some . 1' A' ~ J ' ' ' ' ' ' ' ~J 6~]tD C1lhCDt 1~ A I ~t them .{A ~ ~ A ~ it ~rn7JohiP J~1Vvvv ~"V~ovv En ail, ' ~ Hind VJ VlVvllV, I `' JO ~ ~ ~ ~j J TV r. ~vw~vv {art hunt Girl cry A 1 -rid \~A4~ A4~ ,, ~ ^^il, ..., Ai are not simultaneously satis- fiable in any clomain). Now, if the formulas of each finite subset of {Ao' A1, A2, ...) are simultaneously satisfiable in a respective domain, then the second alternative just above is exclucled, so all the formulas are simul- taneously satisf able (this result is called "compactness"), indeed in the domain {0, i, 2, ...} (the Lowenheim-Skolem theorem). Skolem in (1920), in addition to closing up a gap in Lowen

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.. KURT GODEL 141 helm's (1915) reasoning, added the case of infinitely many formulas. These satisfiability results, which are coupled with the completeness theorem in Godel's treatment, have surprising consequences in certain cases when we aim to use a collection of formulas as axioms to characterize a mathematical system of objects. In doing so, the formulas are not to be logical axioms, but rather mathematical axioms intended to be true, for a given domain D and assignment of predicates over D to the predicate symbols, exactly when D and the predicates have the structure we want the system to have. To make the evaluation process apply as intended, ~ shall suppose the ax- ioms to be closed, that is, to have no free variables. In using the symbolism of the predicate calculus to for- muiate mathematical axioms, we usually want to employ a predicate symbol E(a,b) intended to express a = b, and usually written a=b. Then, for our evaluation process we are only interested in assignments that give E(a,b) the value a= b, that is, that make E(a,b) true exactly when a and b have the same member of D as value. Adding some appropriate axioms to the predicate calculus for this case (Kleene 1952, top 399), we get the f rst-order predicate calculus with equality. Godel of- fered supplementary reasoning that adapted his treatment for the predicate calculus to the predicate calculus with equality, with "the domain {0, I, 2, ...~" being replaced in his conclusions by "~0, I, 2, ...) or a non-empty finite domain". Cantor in (~874) established that the set of all the subsets of the natural numbers {0, I, 2, ...) (or the set of the sets of natural numbers) is more numerous, or has a greater cardinal number, than the set of the natural numbers. To explain this, ~ review some notions of Cantor's theory of sets. He wrote (~895, 4814: "By a 'set' we understand any collection M of definite well-distinguished objects m of our perception or our thought (which are called the 'elements' for 'members'] of M)

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142 BIOGRAPHICAL MEMOIRS into a whole." A set N is a subset of a set M if each member of N is a member of M. For example, the set {0, 1, 2} with the three members shown has the following ~ (= 23) subsets: {0, I, 2), {l, 2), {0, 2), {0, i}, {0}, Ail, {2), ~ }. Cantor showocl that there is no way of pairing all the sets of natural numbers (that is, all the subsets of the natural numbers) with the natural numbers, so that one subset is paired with 0, another with I, still another with 2, and so on, with every natural number used exactly once. Sets have the same cardinal number if they can be thus paired with each other, or put into a "one-to-one correspondence". Denoting the cardinal number of the nat- ural numbers by R0, anc! adopting No as a notation for the cardinal of the sets of natural numbers, thus No ~ 80. But the natural numbers can be paired with a subset of the sets of natural numbers (incleed with the unit sets I,O'J, ISIS, {2i, ... having one member each), so we write No ~ R0. Sets that are either finite or have the cardinal He are called countable; other sets, uncountable. The real numbers (corre- sponding to the points on a line) have the same cardinal No as the sets of natural numbers. ~ have been tacitly assuming for the first-orcl~er predicate calculus (without or with equality) that only a countable col- lection of variables and of predicate symbols is allowed. This entails that only a countable collection of formulas exists. Now suppose that we want to write a list of formulas A,,, ..., An or Ao' Al, A2, ... in the first-order predicate calculus with equality to serve as axioms characterizing the sets in some version of Cantor's theory of sets. Presuming that the axioms are satisfied simultaneously in some domain D (the "sets" in that version of Cantor's set theory) by some assign ment (the on-e understood in his theory), it follows by the Lowenheim-Skolem theorem that they are also satisfiable in the countable domain 10, i, 2, ...~! (It is evident that they are not satisfiable in a finite domain.) That is, one can so interpret

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. KURT GODEL 143 the axioms that the range of the variables in them constitutes a countable collection, contradicting the theorem of Cantor by which the subsets of {0, I, 2, ...} (which are among the sets for his theory) constitute an uncountable collection. This is Skolem's paradox (19231. It is not a direct contradiction; it only shows that we have failecI by our axioms to characterize the system of all the sets for Cantor's theory, as we wished to do. Suppose instead that we want a list of formulas Ao, Al, A2, in the first-order predicate calculus with equality to serve as axioms characterizing the system of the natural numbers 0, I, 2, .... Skolem in (1933, 1934) showed that we cannot succeed in this wish. He constructed so-called "non-standarct moclels of arithmetic", mathematical systems satisfying all the axioms An, Al, As, ... (or indeecI all the formulas that are true in the arithmetic of the natural numbers) but with a clifferent structure (as the mathematicians say, not isomorphic to the nat- ural numbers). In fact, as seems to have been noticec! first by Henkin in (1947), the existence of non-stanclarct models of arithmetic is an immediate consequence of the compactness part of Goclel's completeness theorem for the predicate cal- culus with equality. Before giving Henkin's argument, I observe that we may enlarge the class of formulas for the first-order predicate calculus with equality by allowing individual symbols i, j, k, .... which for any assignment in a domain D are given members of D as their values, and function symbols f, g, h, ..., where, for example, if f is a symbol for a two-place function, its inter- pretation in any assignment is as a function of two variables ranging over D and taking values in D. Examples that come to mincI for systems of axioms for the natural numbers are O as an incTiviclual symbol (with the number O as its stanciard interpretation), ' as a one-place function symbol (to be inter- pretect by + I), and + and x as two-place function symbols

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44 BIOGRAPHICAL MEMOIRS (for addition and multiplication). Such additions to the sym- bolism are not essential. We couIcI equivalently use predicate symbols Z(a), S(a,b), A(a,b,c), and M(a,b,c), where Z(a) is taken as true exactly when the value a of a is O; S(a,b) when a+ ~ = b; A(a,b,c) when a+ b = c; anti M(a,b,c) when ab = c. Here "equivalently" means that whatever can be expressed using the indiviclual and function symbols can be para- phrased using the predicate symbols (but at a considerable loss of convenience). This is shown in Hilbert and Bernays (1934, 460 95.) and Kleene (1952, 741. Now take the proposed list of axioms Ao' Al, A2, ..., which are true under the interpretation by the system of the natural numbers. I shall assume they include (or if necessary add to them) the axioms Vxfx'=O) and Vx~y~x'=y' ~ x=y). Now consider instead the list Ao' ,i = 0, Al, pi = 0', A2, ,i = 0", ... where i is a new individual symbol. Each finite subset of these formulas is true under the intencled interpretation of the oIct symbols and interpreting i by a natural number n for which ~i= 0`n' (with n accents on the 0) is not in the subset. So by compactness, Ao' ~i= 0, Al, ~i= 0', A2, ~i= 0", ... are simultaneously satisfiable. It is easy to see that the satisfying system is isomorphic to one in which 0, i, 2, ... are the values of 0, 0', 0", ... and the value of i is not a natural number a non-stanciard mocle! of arithmetic. These illustrations will suggest the power of Goclel's com- pleteness theorem (1930a) with its corollaries as a too} in studying the possibilities for axiomatically founding various mathematical theories. Actually, not only was the Lowenheim-Skolem theorem around earlier than 1930, but it has been noticed in retro- spect that the completeness of the first-order predicate cal- culus can be derived as an easy consequence of Skolem (19231. Nevertheless, the possibility was overIookecl by Sko- lem himself; indeed the completeness problem was first for

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.. KURT GODEL 169 Quine, W. V. 1978. Kurt Godel ~ 1906 -19781. Year Book Am. Philos. Soc.: 81-84. (On p. 84, for "John von Neumann" read "Julian Schwinger".) Rautenberg, W. 1968. Die Unabhangigkeit der Kontinuumhy- pothese Problematik und Diskussion. Math. in der Schule, 6:18-37. Reinhardt, W. N. 1974. Remarks on reflection principles, large car- dinals, and elementary embeddings. In: Axiomatic Set Theory; Proc. Symposia Pure Math., XIII ~ July 10-August 5, 1967), Part II, ed. T. J. Jech, pp. 187-205. Providence, R.I.: American Mathematical Society. Robinson, A. 1974. Non-Standard Analysis, 2d ed. Amsterdam: North-Holland. Rosser, I. B. 1936. Extensions of some theorems of Godel and Church. ). Symb. Logic, 1: 87-91. Russell, B. 1908. Mathematical logic as based on the theory of types. Am. J. Math., 30:222-262. Saracino, D. H. and V. B. Weispfennig, eds. 1975. Model Theory and Algebra. A Memorial Tribute to Abraham Robinson. Berlin, Heidel- berg, and New York: Springer (Lecture Notes in Mathematics, 498~. Schilpp, P. A., ed. 1944. The Philosophy of Bertrand Russell. Evanston and Chicago, Ill.: Northwestern University Press. Schilpp, P. A., ed. 1949. Albert Einstein, Philosopher-Scientist. Evans- ton, Ill.: Northwestern University Press. (German edition, Albert Einstein als Philosoph und Naturforscher. 1955. Stuttgart: Kohl- hammer.) Skolem, T. 1920. Logisch-kombinatorische Untersuchungen uber die Erfullbarkeit oder Beweisbarkeit mathematischer Satze nebst einem Theoreme uber dictate Mengen. Skrifter utgit av Videnskapsselskapet i Kristiania, I. Matematisk-naturvidenskabelig klasse 1920, no. 4. Skolem, T. 1923. Einige Bemerkungen zur axiomatischen Be- grundung der Mengenlehre. In: Wissenschaftliche Vortrage 5. Kong. Skand. Math. Helsingfors 4-7 [uli 1922. Helsin~fors. on. 217-32. Skolem, T. 1933. Uber die Unmoglichkeit einer vollstandigen Charakterisierung der Zahlenreihe mittels eines endlichen Ax- iomensystems. Norsk matematisk forenings skrifter, ser. 2, no. 10: 73-82. - ~ - -' 1 1

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170 BIOGRAPHICAL MEMOIRS Skolem, T. 1934. Uber die Nicht-charakterisierbarkeit der Zahlen- reihe mittels endlich oder abzahlbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen. Fund. Math., 23: 150-61. Spector, C. 1962. Provably recursive functionals of analysis: A con- sistency proof of analysis by an extension of principles formu- lated in current intuitionistic mathematics. In: Recursive Func- tion Theory; Proc. Symposia Pure Math., V (Abril 6-7, 1961J, ed. I. C. E. Dekker, pp. 1-27. Providence, K.1.: American Mathe- matical Society. (Posthumous, with footnotes and some editing ~-r--~~~ - ~~~ ~ -- - ~~~~~~, \ 1 by Kreisel and a postscript by Godel.) Turing, A. M. 1937. On computable numbers, with an application to the Entscheidungsproblem. Proc. London Math. Soc., ser. 2, 42:230-65. (A correction, 43:544-46.) Ulam, S. 1958. John von Neumann, 1903-1957. Bull. Am. Math. Soc., 64, no. 3, pt. 2: 1 - 49. van Heijenoort, I., ed. 1967. From Frege to Godel: a Source Book in Mathematical Logic, 1879-1931. Cambridge, Mass.: Harvard University Press. (The English translations Land introductory notes thereto] of Godel ~ 1930e, 1931 a, 1931 b) and of Frege ~ 1879) in this volume are reprinted in Frege and Godel: Two Fun- damental Texts in Mathematical Logic, 1970.) van Neumann, I. 1927. Zur Hilbertschen Beweistheorie. Math. Zeit., 26:1-46. von Neumann, l. 1951. ETribute to Kurt Godel quoted in] The New York Times, March 15, 1951, 31. BulloE et al., 1 969. ~ , ,, %, . ~ (More fully on pp. (ix)-(x) of von Neumann, I. 1966. Theory of Self-Reproducing Automata (post- humous), ed. and completed by A. W. Burks. Urbana and Lon- don: University of Illinois Press. Wang, H. 1974. From Mathematics to Philosophy. London: Routledge & Kegan Paul; New York: Humanities Press. Wang, H. 1978. Kurt Godel's intellectual development. Math. Intel- ligencer, 1:182 - 84. Wang, H. 1981. Some facts about Kurt Godel. [. Symb. Logic, 46:653-59. Whitehead, A. N. and B. Russell. 1910, 1912, 1913. Principia Math- ematica. 3 vols. Cambridge, U.K.: Cambridge University Press. Zermelo, E. 1908. Untersuchungen uber die Grundlagen der Men- genlehre I . Math. Ann., 65:261-81.

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.. KU RT G O D E L HONORS AND DISTINCTIONS AWARDS AND MEMBERSHIPS 171 Albert Einstein Award, Lewis and Rosa Strauss Memorial Fund (shared with Julian Schwinger), 1951 National Academy of Sciences, Member, 1955 American Academy of Arts and Sciences, Fellow, 1957 American Philosophical Society, Member, 1961 London Mathematical Society, Honorary Member, 1967 Royal Society (London), Foreign Member, 1968 British Academy, Corresponding Fellow, 1972 Institut de France, Corresponding Member, 1972 Academic des Sciences Morales et Politiques, Corresponding Mem- ber, 1972 National Medal of Science, 1975 HONORARY DEGREES D.Litt., Yale University, 1951 Sc.D., Harvard University, 1952 Sc.D., Amherst College, 1967 Sc.V., Rockefeller University, 1972

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172 BIOGRAPHICAL MEMOIRS ANNOTATED BIBLIOGRAPHY I have listed the original items in the order of their authorship by Godel, insofar as I could find information to base this on. Thus 1939b, commu- nicated by Godel on February 14, 1939, is put after 1939a, which is a set of notes by George W. Brown published in 1940 on lectures delivered by Godel in the fall term of 1938-39; and 1934 and 1946, only published in 1965, are in the right order. This has involved using dates of presentation for the eleven items listed from Ergebnisse eines mathematischen Kolloquiums (ea. Karl Menger). Not listed are four brief contributions by Godel to dis- cussions in these Ergebnisse (4:4, 4:6, 4:34~51), 7:6), twenty-seven reviews by Godel in Zentralblatt fur Mathematik und ihrer Grenzgebiete 1931-36, six reviews by Godel in MonatsheftefurMathematik und Physik 1931-33, and the Spanish translations in Moster~n (1981) of all of the papers of Godel ex- cept 1932a, 1932b, and 1933a. (In Dawson (1983), the four Ergebuisse items are cited (just before F1933a] and as t1933], t1933d1, f 19364), as well as the twenty-seven Zentralblatt reviews and the Moster~n (1981) transla- tions, while the six Monatshefte reviews are cited in its Addenda and Cor- rigenda.) The other translations and reprintings of Godel's papers are cited within the items for the originals. Eight of the translations (with no inputs by Godel, as far as I know) are thus cited concisely through their reviews or listings in the journal of Symbolic Logic. Books (not journals) are generally cited through the Reference section. For example, the English translation of 1930a is on pp. 583-91 of the book listed in the References as "van Heijenoort, l., ed. 1967." A work entitled Kurt Godel, Sein Leben und Wirken, W. Schimanovich and P. Weibel, eds., is to be published by Verlag Holder-Pichler-Tempsky, Vi- enna. It will contain some of Godel's works and various biographical and . . uterpretatlve essays. The Association for Symbolic Logic is arranging for the publication in the original (by Oxford University Press, ed. by S. Feferman et al.), and when the original is in German also in English translation, of all of Godel's pub- lished works, with introductory historical notes to them and a biographical introduction and survey. (Volume I ~ 1986) contains Godel's published works up through 1936; the rest will be in Volume II, probably ir~ 1986. A further volume or volumes are projected to contain a selection of un- published material from Godel's Nachlass.) 1930 a. Die Vollstandigkeit der Axiome des logischen Funktionenkalk uls. Monatsh. Math. Phys., 37:349-60. (English bans: van Hei- jenoort, 1967, pp. 583-91, with two comments by Godel, pp.

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.e KURT GODEL 173 510-11; also see Kleene, 1978; reprinted in: Berka and Kreiser, 1971, pp. 283-94.) b. Uber die Vollstandigkeit des Logikkalkuls (talk of 6 Sept. 1930~. Die Naturwissenschaften, 18: 1068. c. FRemarks in] Diskussion zur Grundlegung der Mathematik F7 Sept. 19301. Erkenntnis (1931), 2:147-48. d. Nachtrag Lto the preceding remarks1. Erkenntnis, 2:149-51. (Italian bans: Casari, 1973, pp. 55-57.) e. Einige metamathematische Resultate uber Entscheidungsdefin- itheit und Widerspruchsfreiheit. Anz. Akad. Wiss. Wien, Math.- naturwiss. K1. 67:214-15. (English trans.: van Heijenoort, 1967, pp. 595-96; reprinted in: Berka and Kreiser, 1971, pp. 320- 21.) f. Ein Spezialfall des Entscheidungsproblems der theoretischen Logik. Ergeb. math. Kolloq. (for 1929-30, publ. 1932), 2:27- 28. 1931 a. Uber formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme I. Monatsh. Math. Phys., 38:173-98. (English trans., 1962, see J. Symb. Logic, 30:359-62; also in: Davis, 1965, pp. 5-38 (see J. Symb. Logic, 31:486-891; with a note by Godel, in: van Heijenoort, 1967, pp. 596-616. Italian trans.: Agazzi, 1961, pp. 203-28; Portuguese trans.: Lourenc,o, 1979, pp. 245-90.) b. Uber Vollstandigkeit und Widerspruchsfreiheit. Ergeb. math. Kolloq. (for 22 Jan. 1931, publ. 1932), 3: 12-13. (English trans., with a remark by Godel added to Ftn. 1: van Heijenoort, 1967, pp. 616-17.) c. Eine Eigenschaft der Realisierung des Aussagenkalkuls. Ergeb. math. Kolloq. (for 24 Tune 1931, publ. 1932), 3:20-21. d. Letter to Zermelo, October 12, 1931. In: Grattan-Guinness, 1979, pp. 294-304. Uber Unabhangigkeitsbeweise im Aussagenkalkul. Ergeb. math. Kolloq. (for 2 Dec. 1931, publ. 1933), 4:9-10. 1932 . . a. Uber die metrische Einbettbarkeit der Quadrupel des R3 in Ku- gelflachen. Ergeb. math. Kolloq. (for 18 Feb. 1932, publ. 1933), 4: 16-17.

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174 BIOGRAPHICAL MEMOIRS b. Uber die Waldsche Axiomatik des Zwischenbegriffes. Ergeb. math. Kolloq. (for 18 Feb. 1932, publ. 1933), 4:17-18. c. Zum intuitionistischen Aussagenkalkul. Anz. Akad. Wiss. Wien, Math.-naturwiss. K1. (for 25 Feb. 1932), 69:65-66. (Reprinted, with an opening clause attributing the question to Hahn, in Ergeb. math. Kolloq. (for 1931-32, publ. 1933), 4:40; and in Berka and Kreiser, 1971, p. 186.) d. Zur intuitionistischen Arithmetik und Zahlentheorie. Ergeb. math. Kolloq. (for 28 June 1932, publ.1933),4:34-38. (English trans.: Davis, 1965, pp. 75-81 (see i. Symb. Logic, 31:490-911. Portuguese trans.: Louren~o, 1 979, pp. 359-69.) Fine Interpretation des intuitionistischen Aussagenkalkuls. Er- geb. math. Kolloq. (for 1931-32, publ.1934), 4:39-40. (English trans., 1969, see J. Symb. Logic, 40:498; reprinted in: Berka and Kreiser, 1971, pp. 187-88~. I. Bemerkung uber projektive Abbildungen. Ergeb. math. Kolloq. (for 10 Nov. 1932, publ. 1934), 5: 1. 1933 a. With K. Menger and A. Wald. Diskussion uber koordinatenlose Differentialgeometrie. Ergeb. math. Kolloq. (for 17 May 1933, publ. 1934), 5:25 -26. Zum Entscheidungsproblem des logischen Funktionenkalkuls. Monatsh. Math. Phys. (received 22 June 1933),40:433-43. (For a correction, see Goldfarb, 1981.) 1934 On Undecidable Propositions of Formal Mathematical Systems. Mimeo- graphed notes by S. C. Kleene and J. B. Rosser on lectures at the Institute for Advanced Study, Feb.-May, 1934, 30 pp. (Ex- tensively distributed, deposited in some libraries, and listed in the }. Symb. Logic Bibliography 1:206; printed with correc- tions, emendations, and a Postscriptum, by Godel in Davis 1965, pp. 41-74 (see J. Symb. Logic, 31:489-90~. A relevant Godel letter of 15 Feb. 1965 is quoted there on p. 40, and in Kleene, 1981, pp. 60, 62, and of 23 April 1963 in van Heijenoort 1967, p. 619. Portuguese trans. in Louren~o 1979, pp. 291-358.) 1935 Uber die Lange von Beweisen. Ergeb. math. Kolloq. (for l9 June 1935, with a remark added in the printing 1936),7:23-24. (En

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.. KURT GODEL 175 glish trans.: Davis 1965, pp. 82-83 (see d. Symb. Logic. 31:491). Portuguese trans.: Lourenc,o 1979, pp. 371-75. 1938 a. The consistency of the axiom of choice and of the generalized continuum-hypothesis. Proc. Natl. Acad. Sci. USA (communi- cated 9 Nov. 1938), 24:556 -57. b. The consistency of the generalized continuum-hypothesis. Bull. Am. Math. Soc. (abstract of a talk on 28 Dec. 1938, publ. 1939), 45:93. 1939 a. The Consistency of the Axiom of Choice and of the Generalized Contin- uum-Hypothesis with the Axioms of Set Theory. Notes by G. W. Brown on lectures at the Institute for Advanced Study during the fall term of 1 938-39. Ann. Math. Stud., no. 3. Princeton, N. I.: Princeton U. Press, 1940. (Reprinted 1951 with corrections and three pages of notes by Godel; the seventh and eighth printings, 1966 and 1970, include additional notes and a bibliography. Russian trans. 1948, see }. Symb. Logic. 14: 142.) Consistency-proof for the generalized continuum-hypothesis. Proc. Natl. Acad. Sci. USA (communicated 14 Feb. 1939), 25:220 - 24. (Reprinted in Feigner, 1979; corrections in (1947, Ftn. 23, = Ftn. 24 in the 1964 reprint); also see Kleene 1978 and Wang 1981, Ftn. 7.) 1944 Russell's mathematical logic. In Schilpp (1944, pp. 123-531. (Re- printed, with a prefatory note by Godel, in Benacerraf and Put- nam 1964, pp. 211-32; Italian trans.: 1967, see J. Symb. Logic, 34:313; French trans.: 1969, see i. Symb. Logic, 40:281; re- printed, with Godel's 1964 prefatory note expanded, a refer- ence supplied in Ftn. 7, and Ftn. 50 omitted, in Pears 1972, pp. 1 92-226; Portuguese trans.: Lourenco 1 979, pp. 1 83 -2 1 6.) 1946 Remarks before the Princeton Bicentennial Conference on Prob- lems in Mathematics EDecember 17i, 1946. Plans for publication of the papers presented at the conference fell through, as the conferees learned only much later. When the Davis anthology

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176 BIOGRAPHICAL MEMOIRS (1965) was being planned, Kleene drew the attention of the publisher to this paper of Godel and supplied a copy of the text that had been in his file since 1946, which with Godel's permis- sion (and Godel's addition of a four-line footnote) was then published as Davis (1965, 84-881. (Italian trans.: 1967, see }. Symb. Logic, 34:313; reprinted, with trifling changes in punc- tuation and phrasing, and the substitution of"It follows from the axiom of replacement" for "It can be proved" at the end, in Klibansky 1968, pp. 250 - 53; Portuguese trans.: Lourenco 1979, pp. 377-83.) 1947 What is Cantor's continuum problem? Am. Math. Mon., 54:515- 25; errata, 55:151. (Reprinted, with some revisions, a substan- tial supplement, and a postscript, by Godel, in Benacerraf and Putnam 1964, pp. 258-73; Italian trans.: 1967, see l. Symb. Logic, 34:313; Romanian trans.: Parvu 1974, pp. 317 - 38; Por- tuguese trans.: Lourenco 1979, pp. 217-44; see (1958c) and (19731.) 1949 a. An example of a new type of cosmological solutions of Einstein's field equations of gravitation. Rev. Mod. Phys., 21:447-50. b. A remark about the relationship between relativity theory and idealistic philosophy. In Schilpp (1949, pp. 555-621. (German trans., with some additions by Godel to the footnotes: Schilpp 1955, pp. 406-12.) 1950 Rotating universes in general relativity theory. In: Proc. Int. Cong. Math. (Cambridge, Mass., 1950), vol. 1, pp. 175-81. Providence, R. I.: American Mathematical Society, 1952. 1952 fA popular interview with Godel:] Inexhaustible. The New Yorker, Aug. 23, 1952, pp. 13-15. 1956 Godel expresses regret at Friedberg's intention to study medicine in: The prodigies. Time, March 19, 1956, p. 83.

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.. KURT GODEL 1957 177 Kreisel (1962, pp. 140-42) states some results as having been com- municated to him by Godel in 1957. 1958 a. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica, 12:280-87. (Reprinted in Logica, Stu- dia Paul Bernays Dedicata, Bibliotheque Scientifique no. 24, pp. 76-83. Neuchatel: Griffon, 1959. Russian trans. 1967, see I. Symb. Logic, 35: 323; English trans.: 1 980 With a bibliography of work resultingfrom this paper], J. Philos. Logic, 9:133-42. According to the review of this translation by Feferman, Math. Rev., 81i:3410-11, there was an unpublished earlier English trans., which was revised several times by Godel and "contained a number of further notes which considerably amplified and in some cases corrected both technical and philosophical points." Italian trans.: Cagnoni 1981, pp. 117-23; also see (1961) and Spector ~ 1961).) b. Kreisel (1958, pp. 321-22) attributes the substance of his re- marks 2.1 and 2.3 to Godel. c. A statement by Godel is quoted in Ulam (1958, Ftn. 5, p. 131. 1961 A postscript by Godel is on p. 27 of Spector (1961). 1963-66 c. a. Benacerraf and Putnam (1964), Davis (1965), and van Heijen- oort (1967) include various contributions by Godel to their re- prints and translations of his (1930a), (1931a), (1931b), (1934), ~ 1944), and ~ 19471. b. A letter from Godel is quoted in von Neumann (1966, pp. 55 561. A 1966 greeting by Godel is on p. (viii) of Bulloff et al. (19691. 1967 An extract from a 30 June 1967 letter from Godel is in Rautenberg (1968, p. 20~. 1973 A communication from Godel of October 1973 is quoted in Green- berg ~ 1980, p. 2501.

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178 BIOGRAPHICAL MEMOIRS 1974 a. Communications from Godel are reproduced in Wang (1974, pp. 8-12, 84-88, 186-90, and 324-261. b. Godel contributed a statement to the preface to Robinson (19741. c. Reinhardt (1974, Ftn. 1, p. 189) reports on discussions with Godel. d. A 1974 memorial tribute to Robinson by Godel appears oppo- site the frontispiece of Saracino and Weispfennig (19751. 1976-77 a. Wang (1981), begins with the words, "The text of this article Lbut not the footnotes and section headings] was done together with Godel in 1976 to 1977 and was approved by him at that time." b. The text of Kleene ~ 1978) is composed of communications from Godel of May and June 1977. See Kreisel's review in the Zen- tralblatt ~ 1 979) 40 1: 1 2-1 3.

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