Open determinacy for class games

  • V. Gitman and J. D. Hamkins, “Open determinacy for class games,” in Foundations of Mathematics, Logic at Harvard, Essays in Honor of Hugh Woodin’s 60th Birthday, A. E. Caicedo, J. Cummings, P. Koellner, and P. Larson, Eds., American Mathematical Society, (expected) 2016. (also available as Newton Institute preprint ni15064)  
    @INCOLLECTION{GitmanHamkins2016:OpenDeterminacyForClassGames,
    author = {Victoria Gitman and Joel David Hamkins},
    title = {Open determinacy for class games},
    booktitle = {Foundations of Mathematics, Logic at Harvard, Essays in Honor of Hugh Woodin's 60th Birthday},
    publisher = {American Mathematical Society},
    year = {(expected) 2016},
    editor = {Andr\'es E. Caicedo and James Cummings and Peter Koellner and Paul Larson},
    volume = {},
    number = {},
    series = {Contemporary Mathematics},
    type = {},
    chapter = {},
    pages = {},
    address = {},
    edition = {},
    month = {},
    note = {also available as Newton Institute preprint ni15064},
    url = {http://jdh.hamkins.org/open-determinacy-for-class-games},
    eprint = {1509.01099},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    abstract = {},
    keywords = {},
    }

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: