  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 G\"odel-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 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 elementary
transfinite recursion ETR 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.
