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?
(In reply to Solution
I don't understand. What is "not well-defined" about the "set of all sets" or the "set of all finite sets"? In other words, how do I distinguish a well-defined set from one that isn't? Is "the set of all integers" well-defined? How about the "set of all subsets of all integers"? ..."the set of all sets of things other than integers"? ..."the set of all sets whose members are not sets"? Where do you draw the line?
To me a well-defined set is one where I can look at any object, and the definition of the set, and decide unambiguously whether or not the object is a member of the set. As such, all the examples I've set forth above are well-defined.