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arXiv · 2406.03801

The behavior of higher proof theory I: Case $Σ^1_2$

Hanul Jeon
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Abstract

Walsh [MR4525964, Zbl 1569.03151] has shown that comparing proof-theoretic ordinals is equivalent to comparing $Π^1_1$-consequence comparison and $Π^1_1$-reflection comparison, all modulo true $Σ^1_1$-sentences. In this paper, we prove the analogous result for $Σ^1_2$-consequences modulo true $Π^1_2$-sentences, that is, the equivalence between $Σ^1_2$-proof-theoretic ordinal comparison, $Σ^1_2$-consequence comparison, and $Σ^1_2$-reflection comparison, all modulo true $Π^1_2$-sentences. We also examine the connection between $Σ^1_2$-proof-theoretic ordinal and $Σ^1_2$-analogue of the robust reflection rank in Pakhomov-Walsh [MR4362917, Zbl 1511.03018]

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Hanul Jeon. 2026-02-08. The behavior of higher proof theory I: Case $Σ^1_2$. https://arxiv.org/abs/2406.03801

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