The exact consistency strength of the generic absoluteness for the universally Baire sets

Abstract

Abstract A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. $\mathsf {Sealing}$ is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by forcing. The $\mathsf {Largest\ Suslin\ Axiom}$ ( $\mathsf {LSA}$ ) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let $\mathsf {LSA-over-uB}$ be the statement that in all (set) generic extensions there is a model of $\mathsf {LSA}$ whose Suslin, co-Suslin sets are the universally Baire sets. We show that over some mild large cardinal theory, $\mathsf {Sealing}$ is equiconsistent with $\mathsf {LSA-over-uB}$ . In fact, we isolate an exact large cardinal theory that is equiconsistent with both (see Definition 2.7). As a consequence, we obtain that $\mathsf {Sealing}$ is weaker than the theory ‘ $\mathsf {ZFC} +$ there is a Woodin cardinal which is a limit of Woodin cardinals’. A variation of $\mathsf {Sealing}$ , called $\mathsf {Tower\ Sealing}$ , is also shown to be equiconsistent with $\mathsf {Sealing}$ over the same large cardinal theory. The result is proven via Woodin’s $\mathsf {Core\ Model\ Induction}$ technique and is essentially the ultimate equiconsistency that can be proven via the current interpretation of $\mathsf {CMI}$ as explained in the paper.