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 numerous situations in which a game of metagame might never complete:-

what happens if the chosen game is drawn - not defined only completes when there is a winner which there will not always be.

what happens if a player takes infinitly long to make his move - not defined doesn't end the game hence the game continues infinetly.

wouldn't be supprised if there are other situations in which metagame does not conclude.

so it's clear that given the rules metagame may or may not conclude depending on how it is played.

this ends the infinate loop problem as the game is obviously not conclusively finite and hence can definitively not be chosen as a choice in metagame. But this still doesn't make metagame definitively finite because the game inherrently does not cover all the situations that would cause the game not to end.

so to answer the is metagame finite or infinite question in the problem, metagame is a game who's rules allow for both finite and infinite games to be played without breaking those rules.

so infact the answer to the question

is Metagame finite or infinite? yes