A characterization of ordinal analysis

James Walsh

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arXiv · 2112.04980

A characterization of ordinal analysis

James Walsh

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Abstract

Ordinal analysis induces a partition of $\Sigma^1_1$-definable and $\Pi^1_1$-sound theories whereby two theories are equivalent if they have the same proof-theoretic ordinal. We show that no equivalence relation $\equiv$ is finer than the ordinal analysis partition if both: (1) $T\equiv U$ whenever $T$ and $U$ prove the same $\Pi^1_1$ sentences; (2) $T\equiv T+U$ for every set $U$ of true $\Sigma^1_1$ sentences. In fact, no such equivalence relation makes a single distinction that the ordinal analysis partition does not make.

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James Walsh. 2021-12-09. A characterization of ordinal analysis. https://arxiv.org/abs/2112.04980

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