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An addendum to “The work of Kurt Gödel”1

Published online by Cambridge University Press:  12 March 2014

Stephen C. Kleene
Stephen C. Kleene*
Affiliation:
University of Wisconsin, Madison, WI 53706
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Extract

Gödel has called to my attention that p. 773 is misleading in regard to the discovery of the finite axiomatization and its place in his proof of the consistency of GCH. For the version in [1940], as he says on p. 1, “The system Σ of axioms for set theory which we adopt [a finite one] … is essentially due to P. Bernays …”. However, it is not at all necessary to use a finite axiom system. Gödel considers the more suggestive proof to be the one in [1939], which uses infinitely many axioms.

His main achievement regarding the consistency of GCH, he says, really is that he first introduced the concept of constructible sets into set theory defining it as in [1939], proved that the axioms of set theory (including the axiom of choice) hold for it, and conjectured that the continuum hypothesis also will hold. He told these things to von Neumann during his stay at Princeton in 1935. The discovery of the proof of this conjecture On the basis of his definition is not too difficult. Gödel gave the proof (also for GCH) not until three years later because he had fallen ill in the meantime. This proof was using a submodel of the constructible sets in the lowest case countable, similar to the one commonly given today.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 43 , Issue 3 , September 1978 , pp. 613
Copyright
Copyright © Association for Symbolic Logic 1978

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Footnotes

1

Stephen C. Kleene, The work of Kurt Gödel, this Journal, vol. 41 (1976), pp. 761–778. Errata: p. 761, line 4 of text, for “of” read “on”; p. 769, line 8, for “[1936]” read “[1943]”.

References

3 Bernays, Paul, A system of axiomatic set theory—Part I, this Journal, vol. 2 (1937), pp. 65–77Google Scholar.

4 The error on p. 773 lines 19–20 is not original with me. In 1975, instead of rereading [1940] from which I learned the subject, or better reading [1939], I used material I had just taught in class from a text by a set-theorist which states, incorrectly, “Gödel was the first to realize that the Comprehension Schema can be replaced by a finite number of its instances …”.

5 Gödel has discussed extensively the conceptual framework of his work in letters and personal communications published in pp. 9–12, 84–86, 186, 189–190, 324–326 of Wang, Hao, From mathematics to philosophy, Routledge & Kegan Paul, London, Humanities Press, New York, 1974, xiv + 428 pp.Google Scholar Also comments by Gödel are in Davis [1965], van Heijenoort [1967] and Robinson, Abraham, Non-standard analysis, 2nd ed., North-Holland, Amsterdam, 1974, xii + 293 pp.Google Scholar