An ordinal analysis of a single stable ordinal

TL;DR

This paper presents an ordinal analysis of a set theory extended with an axiom asserting the existence of a transitive set that is elementarily equivalent to the universe for Σ₁ formulas.

Contribution

It provides a novel ordinal analysis for a set theory with a specific reflection axiom involving a transitive set.

Findings

Ordinal analysis of the extended set theory

Characterization of the proof-theoretic strength

Insights into the structure of models with reflection axioms

Abstract

In this paper we give an ordinal analysis of a set theory extending ${\sf KP}\ell^{r}$ with an axiom stating that `there exists a transitive set $M$ such that $M\prec_{\Sigma_{1}}V$'.