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 . 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 . Player II wins otherwise.
For more information about these kinds of games, you may refer to this wikipedia entry: http://en.wikipedia.org/wiki/Determinacy.