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.