A Note on a Theorem of Apter

TL;DR

This paper explores the implications of certain set-theoretic assumptions, demonstrating how the consistency of one set of axioms leads to the consistency of another, more specific set of axioms involving large cardinals and regularity.

Contribution

It establishes a new consistency implication linking $ ext{ZF} + ext{AD}_ ext{R}$ with the existence of a least strongly regular and measurable cardinal under certain conditions.

Findings

Consistency of $ ext{ZF} + ext{AD}_ ext{R} + ext{``} heta$ is measurable$"$ implies the consistency of a model with a least strongly regular and measurable cardinal.

All uncountable cardinals below $ heta$ are of countable cofinality in the resulting model.

Abstract

We show that the consistency of $\mathrm{ZF} + \mathrm{AD}_{\mathbb{R}} + ``\Theta$ is measurable$"$ implies the consistency of $\mathrm{ZF} +``\Theta$ is the least strongly regular cardinal and the least measurable cardinal$"$ + $``$all uncountable cardinals below $\Theta$ are of countable cofinality$"$.