Well-foundedness proof for ${\Pi }_1^1$-reflection

Abstract

In the lecture notes it is shown that an ordinal ${\psi }_{\Omega }({\epsilon }_{{\mathbb{S}}^{+}+1})$ is an upper bound for the proof-theoretic ordinal of a set theory $\mathsf{KP}\omega +(M{\prec }_{{\Sigma }_1}V)$. In this note we show that $\mathsf{KP}\omega +(M{\prec }_{{\Sigma }_1}V)$ proves the well-foundedness up to ${\psi }_{\Omega }({\omega }_n({\mathbb{S}}^{+}+1))$ for each $n$.

arXiv admin note: substantial text overlap with arXiv:2211.08619, arXiv:2304.00246; text overlap with arXiv:2208.12944