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

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 ${\Sigma }_2^1$-consequences modulo true ${\Pi }_2^1$-sentences, that is, the equivalence between ${\Sigma }_2^1$-proof-theoretic ordinal comparison, ${\Sigma }_2^1$-consequence comparison, and ${\Sigma }_2^1$-reflection comparison, all modulo true ${\Pi }_2^1$-sentences. We also examine the connection between ${\Sigma }_2^1$-proof-theoretic ordinal and ${\Sigma }_2^1$-analogue of the robust reflection rank in Pakhomov-Walsh [MR4362917, Zbl 1511.03018]

33 pages, first revision