{"title":"SEALING OF THE UNIVERSALLY BAIRE SETS","authors":"G. Sargsyan, Nam Trang","doi":"10.1017/bsl.2021.29","DOIUrl":null,"url":null,"abstract":"Abstract A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. \n${\\sf Sealing}$\n 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 \n${\\sf Largest\\ Suslin\\ Axiom}$\n ( \n${\\sf LSA}$\n ) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable surjections. Let \n${\\sf LSA}$\n - \n${\\sf over}$\n - \n${\\sf uB}$\n be the statement that in all (set) generic extensions there is a model of \n$\\sf {LSA}$\n whose Suslin, co-Suslin sets are the universally Baire sets. We outline the proof that over some mild large cardinal theory, \n$\\sf {Sealing}$\n is equiconsistent with \n$\\sf {LSA}$\n - \n$\\sf {over}$\n - \n$\\sf {uB}$\n . 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 \n$\\sf {Sealing}$\n is weaker than the theory “ \n$\\sf {ZFC}$\n + there is a Woodin cardinal which is a limit of Woodin cardinals.” This significantly improves upon the earlier consistency proof of \n$\\sf {Sealing}$\n by Woodin. A variation of \n$\\sf {Sealing}$\n , called \n$\\sf {Tower \\ Sealing}$\n , is also shown to be equiconsistent with \n$\\sf {Sealing}$\n 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 \n$\\sf {Sealing}$\n 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 \n$\\sf {Sealing}$\n 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 \n$\\sf {LSA}$\n - \n$\\sf {over}$\n - \n$\\sf {uB}$\n is not equivalent to \n$\\sf {Sealing}$\n .","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2021-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Bulletin of Symbolic Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/bsl.2021.29","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
Abstract
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
$\sf {Sealing}$
.