$omega_{1}$ under $Pi_{1}$-Collection

TL;DR

This paper establishes a proof-theoretic bound on certain countable ordinals in a specific set theory with collection principles, advancing understanding of the set-theoretic universe's structure.

Contribution

It provides a new proof-theoretic bound for Sigma_2-definable countable ordinals within Kripke-Platek set theory with Pi_1-Collection and omega_1.

Findings

Bound on Sigma_2-definable countable ordinals established

Enhanced understanding of Kripke-Platek set theory with Pi_1-Collection

Implications for the structure of the set-theoretic universe

Abstract

We describe a proof-theoretic bound on $Sigma_{2}$-definable countable ordinals in Kripke-Platek set theory with $Pi_{1}$-Collection and the existence of $omega_{1}$.