# On supercompactness of $\omega_1$

## TL;DR

This paper explores the implications of $oldsymbol{ ext{supercompactness of }oldsymbol{ ext{$oldsymbol{ ext{omega}_1}$}}}$ under $ ext{ZF}$, revealing connections to choice principles, determinacy axioms, and regularity properties of sets.

## Contribution

It establishes new links between supercompactness of $ ext{omega}_1$, choice axioms, determinacy, and regularity properties, under $ ext{ZF}$ without the Axiom of Choice.

## Key findings

- $ ext{DC}$ follows from $ ext{omega}_1$ is supercompact
- $ ext{AD}^+$ is equivalent to $ ext{AD}_ ext{R}$ under supercompactness
- Supercompactness implies all sets in the Chang model have regularity properties

## Abstract

This paper studies structural consequences of supercompactness of $\omega_1$ under $\sf{ZF}$. We show that the Axiom of Dependent Choice $(\sf{DC})$ follows from "$\omega_1$ is supercompact". "$\omega_1$ is supercompact" also implies that $\sf{AD}^+$, a strengthening of the Axiom of Determinacy $(\sf{AD})$, is equivalent to $\sf{AD}_\mathbb{R}$. It is shown that "$\omega_1$ is supercompact" does not imply $\sf{AD}$. The most one can hope for is Suslin co-Suslin determinacy. We show that this follows from "$\omega_1$ is supercompact" and Hod Pair Capturing $(\sf{HPC})$, an inner-model theoretic hypothesis that imposes certain smallness conditions on the universe of sets. "$\omega_1$ is supercompact" on its own implies that every Suslin co-Suslin set is the projection of a determined (in fact, homogenously Suslin) set. "$\omega_1$ is supercompact" also implies all sets in the Chang model have all the usual regularity properties, like Lebesgue measurability and the Baire property.