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?
If "Metagame" was defined so that the first player could pick any finite game EXCEPT METAGAME, then it would be finite.
If the first player has no restrictions, then he can pick METAGAME... and the game becomes infinite, and thus he couldn't have chosen it!
I think the title of the problem is well chosen, and could even have a "and viceversa" appended, except for the alliterative difficulty. ;-)