>>14383555extensionality: games are the same if they have the same available moves
pairing: the game where you choose one of two game states (possibly identical)
union: the game where you take two moves
empty set: the game where you lose
infinity: the game where you can pick a number of nim sticks to put down in a row and your opponent plays first (normal play)
power set: the game where you pick your opponent's available moves
replacement: making one game from another by remapping the available moves
regularity: if the game is not over yet, then one of the moves will take you to a state from which the next player cannot move back to one of the states you could have moved to from the first move. combined with the other game construction techniques, this eliminates the possibility of infinitely looping games.
axiom of choice: if the game is not over yet, and it is impossible for your opponent to maneuver to the same game state on the second move from two different moves made by you on the current move, then there is a game where each of your moves has been replaced by exactly one of the moves your opponent could have moved to on his turn
