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

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 $<\lambda$-strong cardinals.'' The ${\Sigma }_1^2$ reflection property of our model is proved using genericity iterations as used by Neeman and Steel.