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?

There are two aspects to this question, first of all, the rules of the game, and then the length of the game.  Taking this into account, one can argue that Metagame is both finite and infinite.  Finite in that it has a definite end, yet infinite in the possibilities for reaching that end, because a limitless number of finite two person games can be invented, which would meet the requirements of Metagame.

Because it is both finite and infinite, it cannot be included in the list of games selected from, because they must be totally finite (in the length AND rules), and therefore cannot be chosen.  This prevents the endless loop that seems, at first, so apparent.

Posted by Bryan on 2005-04-14 19:36:00