Determinacy from strong compactness of $\omega_1$

Abstract

In the absence of the Axiom of Choice, the "small" cardinal $\omega_1$ can exhibit properties more usually associated with large cardinals, such as strong compactness and supercompactness. For a local version of strong compactness, we say that $\omega_1$ is $X$-strongly compact (where $X$ is any set) if there is a fine, countably complete measure on $\mathcal{P}_{\omega_1}(X)$. Working in $\mathsf{ZF} + \mathsf{DC}$, we prove that the $\mathcal{P}(\omega_1)$-strong compactness and $\mathcal{P}(\mathbb{R})$-strong compactness of $\omega_1$ are equiconsistent with $\mathsf{AD}$ and $\mathsf{AD}_\mathbb{R} + \mathsf{DC}$ respectively, where $\mathsf{AD}$ denotes the Axiom of Determinacy and $\mathsf{AD}_\mathbb{R}$ denotes the Axiom of Real Determinacy. The $\mathcal{P}(\mathbb{R})$-supercompactness of $\omega_1$ is shown to be slightly stronger than $\mathsf{AD}_\mathbb{R} + \mathsf{DC}$, but its consistency strength is not computed precisely. An equiconsistency result at the level of $\mathsf{AD}_\mathbb{R}$ without $\mathsf{DC}$ is also obtained.