Wellfoundedness proof with the maximal distinguished set

Abstract

In arXiv:2208.12944 it is shown that an ordinal ${\sup }_{N<\omega }{\psi }_{{\Omega }_1}({\epsilon }_{{\Omega }_{\mathbb{S}+N}+1})$ is an upper bound for the proof-theoretic ordinal of a set theory $\mathsf{KP}{\ell }^r+(M{\prec }_{{\Sigma }_1}V)$. In this paper we show that a second order arithmetic $Σ^{1-}_{2}\mbox{-CA}+Π^{1}_{1}\mbox{-CA}_{0}$ proves the wellfoundedness up to ${\psi }_{{\Omega }_1}({\epsilon }_{{\Omega }_{\mathbb{S}+N+1}})$ for each $N$. It is easy to interpret $Σ^{1-}_{2}\mbox{-CA}+Π^{1}_{1}\mbox{-CA}_{0}$ in $\mathsf{KP}{\ell }^r+(M{\prec }_{{\Sigma }_1}V)$.

arXiv admin note: text overlap with arXiv:1506.05280, arXiv:2112.09871, arXiv:2208.12944