Biographical Memoirs Volume 56 (1987) / Chapter Skim
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Kurt Godel
Kurt Godel
Pages 134-179
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as second to none among logicians of the modern era, beginning with Frege (18791.2 A third fundamental contribution followed a little later ~ ~ 93Xa, ~ 93Sb, ~ 939a, ~ 939b)
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by Kreisel, 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)
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The restricted or f rst-order predicate calculus ("elementary logic") deals with expressions, callecIformulas, constructed, in accordance with states!
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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)
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In one of the stanciard treatments of the classical firstorder 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)
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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.
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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 axioms to be closed, that is, to have no free variables.
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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)
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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 calculus with equality.
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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 retrospect that the completeness of the first-order predicate calculus can be derived as an easy consequence of Skolem (19231.
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, which Hilbert ant! Ackermann do not mention, and was incisive, obtained the compactness, and incluclecI the supplementary argument to make it apply to the predicate calculus with equality.
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and Wang (1981, Footnote 74) , he toict van Neumann of his plan for proving the relative consistency of the axiom of choice anct the continuum hypothesis by use of his concept of "constructible sets".
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GODEL S INCOMPLETENESS THEOREMS (I93la)
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Brouwer argued that classical logic and mathematics go beyond intuition in treating infinite collections as actually existing. As an example, each of the natural numbers 0, I, 2, ...
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We saw above how the first-order predicate calculus, after logical propositions are expressed as formulas in a precisely regulated symbolic language, was organized by the axiomatic-cleductive method. Whiteheact and Russell, and Hilbert, proposed to do the same for mathematics generally, that is for very substantial portions of mathematics short of the paradoxes.
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In proofs in a system obtained by adding mathematical axioms to the logical apparatus of the first-order predicate calculus (or, as we may call them, "de cluctions" by logic from the mathematical axioms) , we are exploring formulas that are true for each domain D ant]
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This would show that mathematics, as it has been cleveloped classically by adjoining the ideal statements to the real ones, is not getting into trouble. Thus Hilbert proposed to give a kind of justification to the cultivation of those parts of classical mathematics that the intuitionists reject.
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Let us substitute in A(x) for the free occurrences of the variable x successively the expressions (called numerals)
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of those symbols, and the finite sequences of such finite sequences) form a countably infinite collection of linguistic objects.
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, by using a slightly more complicated formula than Godel's G replacecI Godel's hypothesis of m-consistency by the hypothesis of simple consistency.
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The fundamental purpose of using formal systems (as a refinement of the axiomatic-deductive method that has come clown to us from Pythagoras and Euclid in the sixth and fourth centuries B.C.) is to remove all uncertainty about what propositions hold in a given mathematical theory.
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For the formalization of the theory of the natural numbers, using Gode} numberings of the formulas and proofs, the algorithms can be for number-theoretic functions and predicates, so the Church-Turing thesis can be appliecl. Gocle} established his theorem for "Principia Mathematica and related systems".
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To recapitulate, by Godel's first incompleteness theorem, as he gave it in (1931a) , none of the familiar formal systems (like that of Principia Mathematical, and by the generalizecT version of the theorem, which Gocle!
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. With the two incompleteness theorems of Gode} (1931a)
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In (1934) , building on a suggestion of Herbrand (see van Heijenoort (1967, 6191)
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incompleteness theorems. GODEL S RELATIVE CONSISTENCY PROOF FOR THE AXIOM OF CHOICE AND FOR THE GENERALIZED CONTINUUM HYPOTHESIS (Rosa, Sub, ~sssa, ~sssby As remarked above, in Cantor's set theory, the set of the sets of natural numbers, ant!
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(We recall the Skolem paradox about such systems of axioms in first-order logic.) As a standard list of axioms for set theory, ~ will take those commonly called the Zermelo-Fracnke!
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Since nothing is used about sets in this reasoning with L that cannot be based on the axioms ZF, it can be converted as follows into a demonstration that if ZF (taken as the formal system with the mathematical axioms of ZF anct the logical axioms and rules of inference of the first-order predicate calculus) is (simply)
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and Cohen have ushered in a whole new era of set theory, in which a host of problems of the consistency or inclependence of various conjectures relative to this or that set of axioms are being investigated by constructing moclels.
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Wang writes (1981, Footnote 9) , "we may conjecture that between 1943 and 1947 a transition occurred from Goclel's concentration on mathematical logic to other theoretical interests which are primarily philosophical....
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One of these was his Josiah Willard Gibbs Lecture, Some Basic Theorems on the Foundations of Mathematics and their Philosophical Implications, which he read from a manuscript to the American Mathematical Society on December 26, 1951 (} was present)
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Uber eine Eigenschaft des Inbegriffes alter reellen algebraischen Zahlen. four.
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1904. Uber die Grundlagen der Logik und der Arithmetik.
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In: Higher Set Theory, Proceedings, Oberwolfach, Germany, April 13-23, 1977, ed.
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1920. Logisch-kombinatorische Untersuchungen uber die Erfullbarkeit oder Beweisbarkeit mathematischer Satze nebst einem Theoreme uber dictate Mengen.
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1934. Uber die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzahlbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen.
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, 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 Member, 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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For example, the English translation of 1930a is on pp. 583-91 of the book listed in the References as "van Heijenoort, l., ed.
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b. Uber die Vollstandigkeit des Logikkalkuls (talk of 6 Sept.
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291-358.) 1935 Uber die Lange von Beweisen.
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1939 a. The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory.
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I.: American Mathematical Society, 1952. 1952 fA popular interview with Godel:]
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1958 a. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes.
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d. A 1974 memorial tribute to Robinson by Godel appears opposite the frontispiece of Saracino and Weispfennig (19751.
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