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?
The answer to the question "is Metagame finite or infinite?" is no.
Quite simply, Metagame is 'other'; neither finite or infinite. It gains this distinction directly through its self-referential nature.
We have already seen it made clear that Metagame can not be finite or infinite - both categorizations leading to paradoxes. So Metagame cannot be chosen by the first player, and even though there are therefore a finite number of moves in Metagame, it still cannot achieve "finite" status because to confer such status would be to simultaneously eradicate it.
My friend is a Transsexual, so is she/he a man or a woman? The answer is obviously dependent on the definition or perhaps even the allowance for a third option. If we define one group to be all games which aren't the other, then we have only two possible types, but one of the groups will include all the fringe cases and exceptions (So Metagame could then be categorized as "infinite" or more appropriately "not finite" even if it doesn't ever allow an infinite number of moves).
If we define both of our groups, we must either do so well enough to account for all paradoxical cases or must allow for an exception group. We could say 'men' are people with XY chromosomes and 'women' are people with XX chromosomes, but then if someone comes along with a chromosome which mutates upon examination we'd have to call this person 'other'. (I realize this has nothing now to do with Transexuals.) If we had instead defined 'men' as people with XY chromosomes and 'women' as everyone else, then the strange cases would then be included in the definition of 'women' even if they sometimes looked like XY.
In this problem, only the definition of 'finite' is critical to being able to play Metagame, and we can only say Metagame itself is not finite.
Posted by Eric
on 2006-01-09 03:38:21