An incompleteness theorem via ordinal analysis

James Walsh

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arXiv · 2109.09678

An incompleteness theorem via ordinal analysis

James Walsh

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Abstract

We present an analogue of G\"{o}del's second incompleteness theorem for systems of second-order arithmetic. Whereas G\"{o}del 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. 2021-09-20. An incompleteness theorem via ordinal analysis. https://arxiv.org/abs/2109.09678

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