The exact strength of generic absoluteness for the universally Baire sets

Abstract

A set of reals is \textit{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 $\mathsf{Largest}\mathsf{Suslin}\mathsf{Axiom}$ ($\mathsf{LSA}$) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let $\mathsf{LSA}-\mathsf{over}-\mathsf{uB}$ 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 $\mathsf{LSA}-\mathsf{over}-\mathsf{uB}$. In fact, we isolate an exact large cardinal theory that is equiconsistent with both (see \rdef{dfn:hod_pm}). 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 $\mathsf{Tower}\mathsf{Sealing}$, is also shown to be equiconsistent with $\mathsf{Sealing}$ over the same large cardinal theory. The result is proven via Woodin's $\mathsf{Core}\mathsf{Model}\mathsf{Induction}$ technique, and is essentially the ultimate equiconsistency that can be proven via the current interpretation of $\mathsf{CMI}$ as explained in the paper.

This is the latest version

References in corpus (1)

• Sealing from Iterability