On ∑1-definable closed unbounded sets

Omer Ben-Neria, Philipp Lücke*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Definable stationary sets, and specifically, ordinal definable ones, play a significant role in the study of canonical inner models of set theory and the class HOD of hereditarily ordinal definable sets. Fixing a certain notion of definability and an uncountable cardinal, one can consider the associated family of definable closed unbounded sets. In this paper, we study the extent to which such families can approximate the full closed unbounded filter and their dependence on the defining complexity. Focusing on closed unbounded subsets of a cardinal κ which are ∑1-definable in parameters from Hκ and ordinal parameters, we show that the ability of such closed unbounded sets to well approximate the closed unbounded filter on κ can highly vary and strongly depends on key properties of the underlying universe of set theory.

Original languageEnglish
JournalCanadian Journal of Mathematics
DOIs
StateAccepted/In press - 2025

Bibliographical note

Publisher Copyright:
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society.

Keywords

  • Definability
  • Jónsson cardinals
  • large cardinals
  • singular cardinals
  • stationary sets

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