Solovay's inaccessible over a weak set theory without choice

TL;DR

This paper investigates the consistency strength of Lebesgue measurability for certain definable sets within a weak, choice-free set theory, establishing results similar to classical theorems under stronger assumptions.

Contribution

It provides a new analysis of Lebesgue measurability for $oldsymbol{ riangle^1_3}$ sets over Zermelo set theory without the Axiom of Choice, extending classical results to a weaker, choiceless context.

Findings

Established an analogue of the Solovay-Shelah theorem in a weak set theory setting.

Determined the consistency strength required for Lebesgue measurability of $oldsymbol{ riangle^1_3}$ sets.

Analyzed the implications of choice-less axioms on classical measure-theoretic results.

Abstract

We study the consistency strength of Lebesgue measurability for $\Sigma^1_3$ sets over Zermelo set theory ($Z$) in a completely choiceless context. We establish a result analogous to the Solovay-Shelah theorem.