= Choice diagonalization of an undetermined game
With the <axiom of choice>, enumerate both players' binary-move <strategies in an infinite game> in length $\mathfrak c$. Each strategy admits $\mathfrak c$ compatible plays. At stage $\alpha$, reserve a fresh play following I's strategy as a loss for I, and a fresh play following II's strategy as a win for I. Avoid all previously reserved plays, which number less than $\mathfrak c$. The collection of the second reservations is a payoff set against which neither player has a <winning strategy in an infinite game>. This uses no regularity assumption on $\mathfrak c$.
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