Choiceless Löwenheim–Skolem Property and Uniform Definability of Grounds

Toshimichi Usuba*

*Corresponding author for this work

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

Abstract

In this paper, without the axiom of choice, we show that if a certain downward Löwenheim–Skolem property holds then all grounds are uniformly definable. We also prove that the axiom of choice is forceable if and only if the universe is a small extension of some transitive model of ZFC.

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
Pages161-179
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 choice
  • Forcing method
  • Set-theoretic geology

ASJC Scopus subject areas

  • Mathematics(all)

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