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AN INCOMPLETENESS THEOREM VIA ORDINAL ANALYSIS

Part of: Proof theory and constructive mathematics

Published online by Cambridge University Press:  12 September 2022

JAMES WALSH
JAMES WALSH*
Affiliation:
SAGE SCHOOL OF PHILOSOPHY CORNELL UNIVERSITY ITHACA, NY 14850, USA
*
E-mail: jameswalsh@cornell.edu
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Abstract

We present an analogue of Gödel’s second incompleteness theorem for systems of second-order arithmetic. Whereas Gödel showed that sufficiently strong theories that are $\Pi ^0_1$-sound and $\Sigma ^0_1$-definable do not prove their own $\Pi ^0_1$-soundness, we prove that sufficiently strong theories that are $\Pi ^1_1$-sound and $\Sigma ^1_1$-definable do not prove their own $\Pi ^1_1$-soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis.

Keywords

proof theorysecond-order arithmeticordinal analysisincompleteness

MSC classification

Primary: 03F15: Recursive ordinals and ordinal notations 03F40: Gödel numberings and issues of incompleteness

Information

Type
Article
Information
The Journal of Symbolic Logic , Volume 89 , Issue 1 , March 2024 , pp. 80 - 96
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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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