The Dennes Room

Event Detail

Fri Mar 6, 2020
60 Evans Hall, 4:10–6 PM
Logic Colloquium
Nam Trang (UC Irvine)
The Strength of Hom_\infty Sealing

A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. Hom_\infty Sealing (or just 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. In some sense, the largest class of sets of reals for which this Shoenfield-type generic absoluteness can hold is the collection of the universally Baire sets.

The Largest Suslin Axiom (LSA) is a strong determinacy axiom that asserts the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let LSA − over − uB be the statement that in all (set) generic extensions there is a model of LSA whose Suslin, co-Suslin sets are the universallyBaire sets.

We prove that modulo a mild large cardinal hypothesis, Sealing is equiconsistent with LSA-over-UB. A consequence of this is that Sealing is consistent relative to PFA or the existence of a Woodin cardinal which is a limit of Woodin cardinals. We hope to outline some ideas involved in the proof as well as explain why the consistency strength of Hom_\infty sealing being low is a hurdle for inner model theory. This is joint work with Grigor Sargsyan.