Determinacy from strong compactness of $\omega_1$

TL;DR

This paper explores how certain large cardinal-like properties of  in the absence of Choice relate to determinacy axioms, establishing equiconsistency results with  being strongly compact or supercompact under these axioms.

Contribution

It establishes the equiconsistency of 's strong compactness and supercompactness with determinacy axioms in  + C, linking set-theoretic properties to determinacy hypotheses.

Findings

's -strong compactness is equiconsistent with .

's --strong compactness is equiconsistent with _.

's --supercompactness is slightly stronger than _ + C.

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.