Open determinacy for class games

[bibtex key=GitmanHamkins2016:OpenDeterminacyForClassGames]

Abstract. The principle of open determinacy for class games — two-player games of perfect information with plays of length $\omega$, where the moves are chosen from a possibly proper class, such as games on the ordinals — is not provable in Zermelo-Fraenkel set theory ZFC or Godel-Bernays set theory GBC, if these theories are consistent, because provably in ZFC there is a definable open proper class game with no definable winning strategy. In fact, the principle of open determinacy and even merely clopen determinacy for class games implies Con(ZFC) and iterated instances Con(Con(ZFC)) and more, because it implies that there is a satisfaction class for first-order truth, and indeed a transfinite tower of truth predicates $\text{Tr}_\alpha$ for iterated truth-about-truth, relative to any class parameter. This is perhaps explained, in light of the Tarskian recursive definition of truth, by the more general fact that the principle of clopen determinacy is exactly equivalent over GBC to the principle of transfinite recursion over well-founded class relations. Meanwhile, the principle of open determinacy for class games is provable in the stronger theory GBC$+\Pi^1_1$-comprehension, a proper fragment of Kelley-Morse set theory KM.

See my earlier posts on part of this material:


7 thoughts on “Open determinacy for class games

  1. Pingback: Open determinacy for games on the ordinals is stronger than ZFC, CUNY Logic Workshop, October 2015 | Joel David Hamkins

  2. Pingback: Open determinacy for games on the ordinals, Torino, March 2016 | Joel David Hamkins

  3. Pingback: Open and clopen determinacy for proper class games, VCU MAMLS April 2017 | Joel David Hamkins

  4. Pingback: Weak fragments of ETR have least transitive models | Recursively saturated and rather classless

  5. Pingback: Second-order transfinite recursion is equivalent to Kelley-Morse set theory over GBC | Joel David Hamkins

  6. Pingback: On the strengths of the class forcing theorem and clopen class game determinacy, Prague set theory seminar, January 2018 | Joel David Hamkins

  7. Pingback: Open class determinacy is preserved by forcing | Joel David Hamkins

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