A model of the Axiom of Determinacy in which every set of reals is universally Baire

TL;DR

This paper proves the relative consistency of a strong form of the Axiom of Determinacy where all sets of reals are universally Baire, using advanced inner model techniques and large cardinal assumptions.

Contribution

It establishes the consistency of $ ext{ZF} + ext{AD}_ ext{R} +$ 'every set of reals is universally Baire' relative to certain large cardinal hypotheses, extending previous results.

Findings

Proves the consistency of the universal Baire property for all sets of reals under AD.

Uses derived model construction and genericity iterations to establish reflection properties.

Extends prior work on Suslin sets to universally Baire sets.

Abstract

The consistency of the theory $\mathsf{ZF} + \mathsf{AD}_{\mathbb{R}} + {}$``every set of reals is universally Baire'' is proved relative to $\mathsf{ZFC} + {}$``there is a cardinal that is a limit of Woodin cardinals and of strong cardinals.'' The proof is based on the derived model construction, which was used by Woodin to show that the theory $\mathsf{ZF} + \mathsf{AD}_{\mathbb{R}} + {}$``every set of reals is Suslin'' is consistent relative to $\mathsf{ZFC} + {}$``there is a cardinal $\lambda$ that is a limit of Woodin cardinals and of $\mathord{<}\lambda$-strong cardinals.'' The $\Sigma^2_1$ reflection property of our model is proved using genericity iterations as used by Neeman and Steel.