Power $\Sigma_1$ in Card with two Woodin cardinals

TL;DR

This paper demonstrates that, assuming large cardinals, it is consistent for the power set operation to be definable in the class of all cardinals while also having two Woodin cardinals, addressing a question about the interaction of large cardinals and definability.

Contribution

It establishes the consistency of the power set operation being $oldsymbol{ riangle}_1$-definable in the class of all cardinals alongside the existence of two Woodin cardinals, answering a question posed by V"a"an"anen and Welch.

Findings

Consistency of $oldsymbol{ riangle}_1$-definability of power set with two Woodin cardinals

Extension of large cardinal hypotheses with definability properties

Addresses open question in set theory about cardinal definability and large cardinals

Abstract

V\"a\"an\"anen and Welch asked in the paper "When cardinals determine the power set: inner models and H\"artig quantifier logic" which large cardinals are consistent with the power set operation $x\mapsto P(x)$ being $\Sigma_1$-definable in the predicate Card of all cardinals. We show that, relative to large cardinals, this property is consistent together with the existence of two Woodin cardinals.