Games and ramsey-like cardinals

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Abstract

We generalise the α-Ramsey cardinals introduced in Holy and Schlicht (2018) for cardinals α to arbitrary ordinals α, and answer several questions posed in that paper. In particular, we show that α-Ramseys are downwards absolute to the core model K for all α of uncountable cofinality, that strategic ω-Ramsey cardinals are equiconsistent with remarkable cardinals and that strategic α-Ramsey cardinals are equiconsistent with measurable cardinals for all α > ω. We also show that the n-Ramseys satisfy indescribability properties and use them to provide a game-theoretic characterisation of completely ineffable cardinals, as well as establishing further connections between the α-Ramsey cardinals and the Ramsey-like cardinals introduced in Gitman (2011), Feng (1990), and Sharpe and Welch (2011).

Original languageEnglish
Pages (from-to)408-437
Number of pages30
JournalJournal of Symbolic Logic
Volume84
Issue number1
DOIs
Publication statusPublished - Mar 2019

Keywords

  • completely ineffable cardinals
  • core model
  • games
  • ineffable cardinals
  • large cardinals
  • measurable cardinals
  • Ramsey-like cardinals
  • remarkable cardinals
  • virtually measurable cardinals
  • weakly compact cardinals

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