2021/10/06 by Grigor Sargsyan, Sargsyan, Grigor, Nam Trang +1
Mathematics · Computer Science · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.2110.02725
A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. \sfSealing 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 \sfLargest Suslin Axiom (\sfLSA) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let \sfLSA-over-uB be the statement that in all (set) generic extensions there is a model of \sfLSA whose Suslin, co-Suslin sets are the universally Baire sets. We show that over some mild large cardinal theory, \sfSealing is equiconsistent with \sfLSA-over-uB. In fact, we isolate an exact large cardinal theory that is equiconsistent with both (see \rdefdfn:hodpm). As a consequence, we obtain that \sfSealing is weaker than the theory ``\sfZFC + there is a Woodin cardinal which is a limit of Woodin cardinals". A variation of \sfSealing, called \sfTower Sealing, is also shown to be equiconsistent with \sfSealing over the same large cardinal theory. The result is proven via Woodin's \sfCore Model Induction technique, and is essentially the ultimate equiconsistency that can be proven via the current interpretation of \sfCMI as explained in the paper.