Open determinacy for class games
Victoria Gitman, Joel David Hamkins · Contemporary mathematics - American Mathematical Society · 2017
The principle of open determinacy for class games|two-player games of perfect information with plays of length !, where the moves are cho- sen from a possibly proper class, such as games on the ordinals|is not prov- able in Zermelo-Fraenkel set theory ZFC or Godel-Bernays set theory GBC, if these theories are consistent, because provably in ZFC there is a denable open proper class game with no denable winning strategy. In fact, the principle of open determinacy and even merely clopen determinacy for class games implies Con(ZFC) and iterated instances Con (ZFC) and more, because it implies that there is a satisfaction class for rst-order truth, and indeed a transnite tower of truth predicates Tr for iterated truth-about-truth, relative to any class pa- rameter. This is perhaps explained, in light of the Tarskian recursive denition of truth, by the more general fact that the principle of clopen determinacy is exactly equivalent over GBC to the principle of transnite recursion over well- founded class relations. Meanwhile, the principle of open determinacy for class games is provable in the stronger theory GBC + 1 -comprehension, a proper fragment of Kelley-Morse set theory KM.