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

Abstract

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