Lifting up the proof theory to the countables : Zermelo-Fraenkel set theory

TL;DR

This paper develops a proof-theoretic framework for understanding countable ordinals within ZF set theory by using iterations of Mostowski collapsings, providing bounds on definable countable ordinals.

Contribution

It introduces a novel method to analyze countable ordinals in ZF through iterations of Mostowski collapsings, establishing proof-theoretic bounds.

Findings

Bound on definable countable ordinals in ZF

Use of iterations of Mostowski collapsings to characterize ordinals

Proof-theoretic analysis of set-theoretic structures

Abstract

We describe the countable ordinals in terms of iterations of Mostowski collapsings. This gives a proof-theoretic bound of definable countable ordinals in the Zermelo-Fraenkel's set theory ZF.