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A MATHEMATICAL COMMITMENT WITHOUT COMPUTATIONAL STRENGTH

Part of: Discrete mathematics in relation to computer science Proof theory and constructive mathematics Philosophical aspects of logic and foundations

Published online by Cambridge University Press:  02 July 2021

ANTON FREUND
ANTON FREUND*
Affiliation:
FACHBEREICH MATHEMATIK TECHNICAL UNIVERSITY OF DARMSTADT SCHLOSSGARTENSTR. 7 64289 DARMSTADT, GERMANY E-mail: freund@mathematik.tu-darmstadt.de
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Abstract

We present a new manifestation of Gödel’s second incompleteness theorem and discuss its foundational significance, in particular with respect to Hilbert’s program. Specifically, we consider a proper extension of Peano arithmetic ($\mathbf {PA}$) by a mathematically meaningful axiom scheme that consists of $\Sigma ^0_2$-sentences. These sentences assert that each computably enumerable ($\Sigma ^0_1$-definable without parameters) property of finite binary trees has a finite basis. Since this fact entails the existence of polynomial time algorithms, it is relevant for computer science. On a technical level, our axiom scheme is a variant of an independence result due to Harvey Friedman. At the same time, the meta-mathematical properties of our axiom scheme distinguish it from most known independence results: Due to its logical complexity, our axiom scheme does not add computational strength. The only known method to establish its independence relies on Gödel’s second incompleteness theorem. In contrast, Gödel’s theorem is not needed for typical examples of $\Pi ^0_2$-independence (such as the Paris–Harrington principle), since computational strength provides an extensional invariant on the level of $\Pi ^0_2$-sentences.

Keywords

independencecomputational strengthGödel’s second incompleteness theoremHilbert’s programKruskal’s theorempolynomial-time algorithm

MSC classification

Primary: 03F30: First-order arithmetic and fragments
Secondary: 03F40: Gödel numberings and issues of incompleteness 03A05: Philosophical and critical 68R10: Graph theory (including graph drawing)
Type
Research Article
Information
The Review of Symbolic Logic , Volume 15 , Issue 4 , December 2022 , pp. 880 - 906
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

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