Power ${\Sigma }_1$ in Card with two Woodin cardinals

Abstract

Väänänen and Welch asked in the paper "When cardinals determine the power set: inner models and Härtig 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.

8 pages. Changes this version: improved exposition and discussion of the situation in $M_1$