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SEALING OF THE UNIVERSALLY BAIRE SETS

Part of: Set theory

Published online by Cambridge University Press:  02 July 2021

GRIGOR SARGSYAN  and
NAM TRANG
GRIGOR SARGSYAN
Affiliation:
DEPARTMENT OF MATHEMATICS RUTGERS UNIVERSITYNEW BRUNSWICK, NJ, USAE-mail: gs481@math.rutgers.edu
NAM TRANG
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NORTH TEXASDENTON, TX, USAE-mail: Nam.Trang@unt.edu
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Abstract

A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. ${\sf Sealing}$ 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 set forcings. The ${\sf Largest\ Suslin\ Axiom}$ ( ${\sf LSA}$ ) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable surjections. Let ${\sf LSA}$ - ${\sf over}$ - ${\sf uB}$ be the statement that in all (set) generic extensions there is a model of $\sf {LSA}$ whose Suslin, co-Suslin sets are the universally Baire sets. We outline the proof that over some mild large cardinal theory, $\sf {Sealing}$ is equiconsistent with $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$ . In fact, we isolate an exact theory (in the hierarchy of strategy mice) that is equiconsistent with both (see Definition 3.1). As a consequence, we obtain that $\sf {Sealing}$ is weaker than the theory “ $\sf {ZFC}$ + there is a Woodin cardinal which is a limit of Woodin cardinals.” This significantly improves upon the earlier consistency proof of $\sf {Sealing}$ by Woodin. A variation of $\sf {Sealing}$ , called $\sf {Tower \ Sealing}$ , is also shown to be equiconsistent with $\sf {Sealing}$ over the same large cardinal theory. We also outline the proof that if V has a proper class of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, then $\sf {Sealing}$ holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to the aforementioned equiconsistency result, where it is shown that $\sf {Sealing}$ holds in a generic extension of a certain minimal universe. This theorem is more general in that no minimal assumption is needed. A corollary of this is that $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$ is not equivalent to $\sf {Sealing}$ .

Keywords

mouseinner model theorydescriptive set theoryhod mouse

MSC classification

Primary: 03E15: Descriptive set theory
Secondary: 03E45: Inner models, including constructibility, ordinal definability, and core models 03E60: Determinacy principles
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 27 , Issue 3 , September 2021 , pp. 254 - 266
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Copyright
© 2021, Association for Symbolic Logic

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