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?

I don't see what the big fuss is. Metagame simply is not a finite
game. There is no logical paradox here. In other words,
there is no line of reasoning that begins with the statement, "Assume
Metagame is infinite", and ends with a false statement (like "1=2").