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

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

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]
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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
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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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.

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