An Incompleteness Theorem Via Ordinal Analysis

Journal of Symbolic Logic 89 (1):80-96 (2024)
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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.

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James Walsh
New York University

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References found in this work

Reflection ranks and ordinal analysis.Fedor Pakhomov & James Walsh - 2021 - Journal of Symbolic Logic 86 (4):1350-1384.
Notation systems for infinitary derivations.Wilfried Buchholz - 1991 - Archive for Mathematical Logic 30 (5-6):277-296.
The incompleteness theorems after 70 years.Henryk Kotlarski - 2004 - Annals of Pure and Applied Logic 126 (1-3):125-138.
On the diagonal lemma of Gödel and Carnap.Saeed Salehi - 2020 - Bulletin of Symbolic Logic 26 (1):80-88.

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