## 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.