The Axiom of Choice and indeterminacy

It’s been a while since my last post — I apologize to any devoted fans who have been disappointed.

I’m taking a class on Descriptive Set Theory now.  Here is the proof of an interesting theorem from class: Assuming the Axiom of Choice, there exists an undetermined game.

Here, a game consists of two players, a pruned tree, and a payoff set A.  The players move alternately by picking an immediate extension of the last move.  Player I wins if, after infinitely many moves, they have created an element of A.  Player II wins otherwise.

For more information about these kinds of games, you may refer to this wikipedia entry:

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