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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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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the structure of sentences has been replaced by the more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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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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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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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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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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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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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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Representative terms from entire chapter:
predicate calculus
