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Sealing of the universally Baire sets

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This article outlines the proof that over some mild large cardinal theory, Sealing is equiconsistent with Largest Suslin Axiom (LSA)-over-uB. 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 universally Baire sets.

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16 p.

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Sargsyan, Grigor & Trang, Nam July 2, 2021.

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This article outlines the proof that over some mild large cardinal theory, Sealing is equiconsistent with Largest Suslin Axiom (LSA)-over-uB. 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 universally Baire sets.

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16 p.

Notes

Abstract: A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. 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 Largest Suslin Axiom (LSA) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable surjections. 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 universally Baire sets. We outline the proof that over some mild large cardinal theory, Sealing is equiconsistent with LSA-over-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 Sealing is weaker than the theory “ZFC+there is a Woodin cardinal which is a limit of Woodin cardinals”. This significantly improves upon the earlier consistency proof of Sealing by Woodin. A variation of Sealing, called Tower Sealing, is also shown to be equiconsistent with 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 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 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 LSA-over-uB is not equivalent to Sealing.

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  • Bulletin of Symbolic Logic, 27(3), Cambridge University Press, July 2, 2021, pp. 1-16

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  • Publication Title: Bulletin of Symbolic Logic
  • Volume: 27
  • Issue: 3
  • Peer Reviewed: Yes

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  • July 2, 2021

Added to The UNT Digital Library

  • Oct. 21, 2021, 11:30 a.m.

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  • Nov. 20, 2023, 2:27 p.m.

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Sargsyan, Grigor & Trang, Nam. Sealing of the universally Baire sets, article, July 2, 2021; (https://digital.library.unt.edu/ark:/67531/metadc1852275/: accessed April 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Science.

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