Most two person games are finite; for example, chess has rules that don't allow an infinite game, and tic-tac-toe obviously ends after at most 9 plays.

Let's define a new two person game: the "Metagame". The first player first picks any two person finite game (e.g., chess or tic-tac-toe). Then, the second player sets up the board (or whatever is needed) and makes the first move in that game, and the Metagame winner will be whoever wins that game.

The question: is Metagame finite or infinite?

Since every game you play in Metagame is finite, Metagame must be finite. However, if Metagame is finite, then you can pick Metagame to play. Then, the other player can pick Metagame. Then, you can pick Metagame. You can keep doing this, so Metagame is infinite. It is a paradox!

Posted by Math Man on 2013-03-11 09:54:50