Wellfoundedness proof with the maximal distinguished set

Abstract

In Arai (An ordinal analysis of a single stable ordinal, submitted) it is shown that an ordinal $\sup _{N<\omega }\psi _{\varOmega _{1}}(\varepsilon _{\varOmega _{{\mathbb {S}}+N}+1})$ 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 $\Sigma ^{1-}_{2}{\mathrm {-CA}}+\Pi ^{1}_{1}{\mathrm {-CA}}_{0}$ proves the wellfoundedness up to $\psi _{\varOmega _{1}}(\varepsilon _{\varOmega _{{\mathbb {S}}+N+1}})$ for each N. It is easy to interpret $\Sigma ^{1-}_{2}{\mathrm {-CA}}+\Pi ^{1}_{1}{\mathrm {-CA}}_{0}$ in $\mathsf {KP}\ell ^{r}+(M\prec _{\Sigma _{1}}V)$.