On Supercompactness of ω1

Daisuke Ikegami, Nam Trang

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

This paper studies structural consequences of supercompactness of ω1 under ZF. We show that the Axiom of Dependent Choice (DC) follows from “ ω1 is supercompact”. “ ω1 is supercompact” also implies that AD+, a strengthening of the Axiom of Determinacy (AD), is equivalent to ADR. It is shown that “ ω1 is supercompact” does not imply AD. The most one can hope for is Suslin determinacy. We show that this follows from “ ω1 is supercompact” and Hod Pair Capturing (HPC), an inner-model theoretic hypothesis that imposes certain smallness conditions on the universe of sets. “ ω1 is supercompact” on its own implies that every Suslin set is the projection of a determined (in fact, homogenously Suslin) set. “ ω1 is supercompact” also implies all sets in the Chang model have all the usual regularity properties, like Lebesgue measurability and the Baire property.

Original languageEnglish
Title of host publicationAdvances in Mathematical Logic - Dedicated to the Memory of Professor Gaisi Takeuti, SAML 2018, Selected, Revised Contributions
EditorsToshiyasu Arai, Makoto Kikuchi, Satoru Kuroda, Mitsuhiro Okada, Teruyuki Yorioka
PublisherSpringer
Pages27-45
Number of pages19
ISBN (Print)9789811641725
DOIs
Publication statusPublished - 2021
EventSymposium on Advances in Mathematical Logic, SAML 2018 - Kobe, Japan
Duration: 2018 Sep 182018 Sep 20

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume369
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Conference

ConferenceSymposium on Advances in Mathematical Logic, SAML 2018
Country/TerritoryJapan
CityKobe
Period18/9/1818/9/20

Keywords

  • Axiom of determinacy
  • Large cardinal properties
  • Supercompactness
  • ω

ASJC Scopus subject areas

  • Mathematics(all)

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