This thesis presents my contributions to various aspects of the theory of universally Baire sets. One of these aspects is the smallest inner model containing all reals whose all sets of reals are universally Baire (viz., $L(\mathbb {R})$) and its relation to its inner model $\mathsf {HOD}$. We verify here that $\mathsf {HOD}^{L(\mathbb {R})}$ enjoys a form of local definability inside $L(\mathbb {R})$, further justifying its characterization as a “core model” in $L(\mathbb {R})$. We then study a “bottom-up” construction of more complicated universally Baire sets (more generally, determined sets). This construction allows us to give an “L-like” description of the minimum model of $\mathsf {AD}_{\mathbb {R}} + \mathsf {Cof}(\Theta ) = \Theta $. A consequence of this description is that this minimum model is contained in the Chang-plus model. Our construction, together with Woodin’s work on the Chang-plus model, shows that a proper class of Woodin cardinals which are limits of Woodin cardinals implies the existence of a hod mouse with a measurable limit of Woodin cardinals whose strategy is universally Baire.

Another aspect of the theory of universally Baire sets is the generic absoluteness and maximality associated with them. We include some results concerning generic $\Sigma _1^{H(\omega _2)}$-absoluteness with universally Baire sets as predicates or parameters, as well as generic $\Pi _2^{H(\omega _2)}$-maximality with universally Baire sets as predicates. In the second case, we are led to consider the general question of when a model of an infinitary propositional formula can be added by a stationary-set-preserving poset. We characterize when this happens in terms of a game which is a variant of the Model Existence Game. We then give a sufficient condition for this in terms of generic embeddings.

Abstract taken directly from the thesis

E-mail: obrad@math.ucla.edu

URL: https://hal.science/view/index/docid/5273407