Sealing of the universally baire sets

Bulletin of Symbolic Logic 27 (3):254-266 (2021)
  Copy   BIBTEX

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 ${\sf Largest\ Suslin\ Axiom}$ 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 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 that is equiconsistent with both. 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 $\sf {Sealing}$.

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 92,075

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

A minimal counterexample to universal baireness.Kai Hauser - 1999 - Journal of Symbolic Logic 64 (4):1601-1627.
A Minimal Counterexample To Universal Baireness.Kai Hauser - 1999 - Journal of Symbolic Logic 64 (4):1601-1627.
Extending Baire property by uncountably many sets.Paweł Kawa & Janusz Pawlikowski - 2010 - Journal of Symbolic Logic 75 (3):896-904.
Universally baire sets and generic absoluteness.Trevor M. Wilson - 2017 - Journal of Symbolic Logic 82 (4):1229-1251.
Homogeneously Suslin sets in tame mice.Farmer Schlutzenberg - 2012 - Journal of Symbolic Logic 77 (4):1122-1146.
Characterising subsets of ω1 constructible from a real.P. D. Welch - 1994 - Journal of Symbolic Logic 59 (4):1420 - 1432.
Decomposing baire functions.J. Cichoń, M. Morayne, J. Pawlikowski & S. Solecki - 1991 - Journal of Symbolic Logic 56 (4):1273 - 1283.
Measurable chromatic numbers.Benjamin D. Miller - 2008 - Journal of Symbolic Logic 73 (4):1139-1157.
Trichotomies for Ideals of Compact Sets.É Matheron, S. Solecki & M. Zelený - 2006 - Journal of Symbolic Logic 71 (2):586 - 598.

Analytics

Added to PP
2022-11-14

Downloads
8 (#1,320,049)

6 months
4 (#794,133)

Historical graph of downloads
How can I increase my downloads?

Citations of this work

No citations found.

Add more citations

References found in this work

Inner models in the region of a Woodin limit of Woodin cardinals.Itay Neeman - 2002 - Annals of Pure and Applied Logic 116 (1-3):67-155.
Descriptive inner model theory.Grigor Sargsyan - 2013 - Bulletin of Symbolic Logic 19 (1):1-55.
Core models with more Woodin cardinals.J. R. Steel - 2002 - Journal of Symbolic Logic 67 (3):1197-1226.
Local Kc Constructions.J. R. Steel - 2007 - Journal of Symbolic Logic 72 (3):721 - 737.

Add more references