Games with filters i

Matthew Foreman*, Menachem Magidor, Martin Zeman

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call Welch games. Player II having a winning strategy in the Welch game of length ω on κ is equivalent to weak compactness. Winning the game of length 2κ is equivalent to κ being measurable. We show that for games of intermediate length γ, II winning implies the existence of precipitous ideals with γ-closed, γ-dense trees. The second part shows the first is not vacuous. For each γ between ω and κ+, it gives a model where II wins the games of length γ, but not γ+. The technique also gives models where for all ω1 < γ ≤ κ there are κ-complete, normal, κ+-distributive ideals having dense sets that are γ-closed, but not γ+-closed.

Original languageEnglish
Article number2450003
JournalJournal of Mathematical Logic
Volume24
Issue number3
DOIs
StatePublished - 1 Dec 2024

Bibliographical note

Publisher Copyright:
© 2022 World Scientific Publishing Company.

Keywords

  • Games played with filters
  • measurable cardinals
  • precipitous ideals
  • weakly compact cardinals

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