Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T01:36:55.332Z Has data issue: false hasContentIssue false
  • English
  • Français

THE COLLAPSE OF THE HILBERT PROGRAM: A VARIATION ON THE GÖDELIAN THEME

Part of: Philosophical aspects of logic and foundations History of mathematics and mathematicians

Published online by Cambridge University Press:  23 March 2022

SAUL A. KRIPKE Open the ORCID record for SAUL A. KRIPKE [Opens in a new window]
SAUL A. KRIPKE*
Affiliation:
THE SAUL KRIPKE CENTER AND THE GRADUATE CENTER CITY UNIVERSITY OF NEW YORKNEW YORK, NY, USAE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The Hilbert program was actually a specific approach for proving consistency, a kind of constructive model theory. Quantifiers were supposed to be replaced by ε-terms. εxA(x) was supposed to denote a witness to $\exists xA(x)$ , or something arbitrary if there is none. The Hilbertians claimed that in any proof in a number-theoretic system S, each ε-term can be replaced by a numeral, making each line provable and true. This implies that S must not only be consistent, but also 1-consistent ( ${\Sigma}_1^0$ -correct). Here we show that if the result is supposed to be provable within S, a statement about all ${\Pi}_2^0$ statements that subsumes itself within its own scope must be provable, yielding a contradiction. The result resembles Gödel’s but arises naturally out of the Hilbert program itself.

Keywords

HilbertAnsatz1-consistentΠ02Gödel

MSC classification

Primary: 01A60: 20th century
Secondary: 03A05: Philosophical and critical
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 28 , Issue 3 , September 2022 , pp. 413 - 426
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This paper is based on a transcript of a lecture given at Indiana University on October 15, 2007 in honor of the inauguration of President Michael A. McRobbie. An abstract of another version of this talk (given at the 2008 Winter Meeting of the Association for Symbolic Logic on December 30, 2008) was published in The Bulletin of Symbolic Logic, vol. 15 (2009), no. 2, pp. 229–231.

References

Ackermann, W., Begründung des “tertium non datur” mittels der Hilbertschen Theorie der Widerspruchsfreiheit. Mathematische Annalen, vol. 93 (1924), pp. 1–36.CrossRefGoogle Scholar
Ackermann, W., Zur Widerspruchsfreiheit der Zahlentheorie . Mathematische Annalen , vol. 117 (1940), pp. 162–194.CrossRefGoogle Scholar
Craig, W., On axiomatizability within a system . The Journal of Symbolic Logic , vol. 18 (1953), no. 1, pp. 30–32.CrossRefGoogle Scholar
van Dalen, D., The war of the frogs and the mice, or the crisis of the Mathematische Annalen . The Mathematical Intelligencer , vol. 12 (1990), no. 4, pp. 17–31.CrossRefGoogle Scholar
Dawson, J., Logical Dilemmas: The Life and Work of Kurt Gödel , Metaphysics Research Lab, Stanford University, Stanford, 1997.Google Scholar
Ewald, W., The emergence of first-order logic, The Stanford Encyclopedia of Philosophy (Spring 2019 edition) (E. N. Zalta, editor), 2019. Available at https://plato.stanford.edu/archives/spr2019/entries/logic-firstorder-emergence/.Google Scholar
Feferman, S., Dawson, J. W. Jr., Goldfarb, W., Parsons, C. and Solovay, R. M., Kurt Gödel: Collected Works. Volume III: Unpublished Essays and Lectures , Oxford University Press, New York, 1995.Google Scholar
Feferman, S., Dawson, J. W. Jr., Kleene, S. C., Moore, G. H., Solovay, R. M. and van Heijenoort, J., Kurt Gödel: Collected Works. Volume I: Publications 1929–1936, Oxford University Press, New York, 1986.Google Scholar
Feferman, S., Dawson, J. W. Jr., Kleene, S. C., Moore, G. H., Solovay, R. M. and van Heijenoort, J., Kurt Gödel: Collected Works. Volume II: Publications 1938–1974, Oxford University Press, New York, 1990.Google Scholar
Friedberg, R. M., Two recursively enumerable sets of incomparable degrees of unsolvability. Proceedings of the National Academy of Sciences of the United States of America, vol. 43 (1957), pp. 236–238.CrossRefGoogle ScholarPubMed
Gentzen, G., Die Widerspruchsfreiheit der reinen Zahlentheorie . Mathematische Annalen , vol. 112 (1936), pp. 493–565.CrossRefGoogle Scholar
Gödel, K., Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198. Reprinted and translated as On formally undecidable propositions of Principia Mathematica and related systems I in [8], pp. 144–195.CrossRefGoogle Scholar
Gödel, K., Die Vollständigkeit der Axiome des logischen Funktionenkalküls, Monatshefte für Mathematik und Physik, vol. 37 (1930), pp. 349–360. Reprinted in [8], pp. 102–123.Google Scholar
Gödel, K., Vortrag bei Zilsel. Translated as Lecture at Zilsel’s by C. Parsons in [7], pp. 62–113.Google Scholar
Gödel, K., Russell’s mathematical logic, The Philosophy of Bertrand Russell, vol. 3 (P. A. Schlipp, editor). Reprinted in [9], pp. 119–141.Google Scholar
Gödel, K., Über eine bisher noch nicht benütze Erweiterung des finiten Standpunktes, Dialectica, 1958, pp. 280–287. Reprinted and translated as On a hithero unutilized extension of the finitary standpoint by S. Bauer-Mengelberg and J. van Heijenoort in [9], pp. 217–251.CrossRefGoogle Scholar
Hilbert, D., Über das Unendliche. Mathematische Annalen, vol. 95 (1926), pp. 161–190. Lecture given Münster, June 4, 1925. Reprinted and translated as On the infinite by S. Bauer-Mengelberg in [32], pp. 367–392.CrossRefGoogle Scholar
Hilbert, D. and Ackermann, W., Grundzüge der Theoretischen Logik, Springer, Berlin, 1928.Google Scholar
Hilbert, D. and Bernays, P., Grundlagen der Mathematik, Volume 1, Springer, Berlin, 1934.Google Scholar
Kreisel, G., On the interpretation of non-finitist proofs I . The Journal of Symbolic Logic , vol. 16 (1951), no. 2, pp. 241–267.Google Scholar
Kreisel, G., On the interpretation of non-finitist proofs II . The Journal of Symbolic Logic , vol. 17 (1952), no. 1, pp. 43–58.CrossRefGoogle Scholar
Kreisel, G., A refinement of ω-consistency (abstract). The Journal of Symbolic Logic , vol. 22 (1957), no. 1957, pp. 108–109.Google Scholar
Kreisel, G., Ordinal logics and the characterization of informal notions of proof , Proceedings of the International Congress of Mathematicians, Edinburgh, 14–21 August 1958 (J. A. Todd, editor), Cambridge University Press, Cambridge, 1960, pp. 289–299.Google Scholar
Kripke, S. A., The collapse of the Hilbert program (abstract), this Journal, vol. 15 (2009), no. 2, pp. 229–231.Google Scholar
Kripke, S. A., A model-theoretic approach to Gödel’s theorem , Logical Troubles. Collected Papers , vol. II, Oxford University Press, New York, forthcomingGoogle Scholar
Mints, G., Incomplete proofs and program synthesis (extended abstract), AAAI Technical report SS-02-05, 2002. Available at http://www.aaai.org.Google Scholar
Muchnik, A. A., Negative answer to the problem of reducibility of the theory of algorithms (in Russian). Doklady Akademii Nauk SSSR , vol. 108 (1956), pp. 194–197.Google Scholar
Sieg, W., Hilbert’s programs: 1917–1922, this Journal, vol. 5 (1999), no. 1, pp. 1–44.Google Scholar
Skolem, T., Begründung der elementaren Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbarer Veränderlichen mit unendlichem Ausdehnungsbereich, Skrifter utgit av Videnskapsselskapet i Kristiania. I, Matematisk-naturvidenskabelig klasse 6 (1923), pp. 1–38. Reprinted and translated as The foundations of elementary arithmetic established by means of the recursive mode of thought, without the use of apparent variables ranging over infinite domains by S. Bauer-Mengelberg in [32], pp. 302–333.Google Scholar
Tait, W. W., Constructive reasoning , Logic, Methodology and Philosophy of Science III (B. van Rootselar and J. F. Staal, editors), North-Holland, Amsterdam, 1968, pp. 185–199.CrossRefGoogle Scholar
Tait, W. W., Finitism . The Journal of Philosophy , vol. 78 (1981), pp. 524–546.CrossRefGoogle Scholar
van Heijenoort, J., From Frege to Gödel: Source Book in Mathematical Logic, 1879–1931 , Harvard University Press, Cambridge, 1967.Google Scholar
Wang, H., Reflections on Kurt Gödel , MIT Press, Cambridge, 1987.Google Scholar
Whitehead, A. N. and Russell, B., Principia Mathematica , vols. I–III, Cambridge University Press, Cambridge, 1910, 1912, 1913. Second edition: 1925 (vol. 1) and 1927 (vols. 2 and 3).Google Scholar
Zach, R., The practice of finitism. Epsilon calculus and consistency proofs in Hilbert’s program . Synthese , vol. 137 (2003), pp. 211–259.CrossRefGoogle Scholar