 A game is said to be determined if one or other of the players has a winning strategy. There are many examples of games that are not determined, as Gale and Stewart showed in 1953 using the axiom of choice. This discovery led to important applications of the notion of determinacy in the foundations of set theory (see entry on  large cardinals and determinacy).  Gale and Stewart also proved an important theorem that bears their name: Every well-founded game is determined. It follows that every game of finite length is determined—a fact already known to Zermelo in 1913. (A more precise statement of the Gale-Stewart theorem is this. A game \(G\) is said to be closed if \(\exists\) wins every play of \(G\) in which she hasn’t already lost at any finite position. The theorem states that every closed game is determined. The proof of the theorem is basically easy: Let us call a position winning for \(\forall\) if he has a winning strategy starting from this position. Suppose \(\forall\) does not have a winning strategy in the game, that is, in the beginning the position is not winning for \(\forall\). If the first move is a move of \(\forall\), after his move the position is still not winning for him. If the first move is a move of \(\exists\), she must have a move after which the position is still not winning for \(\forall\), for otherwise the previous position would have been winning for \(\forall\). The game goes on in this way infinitely many moves through positions which are not winning for \(\forall\). Because the game is closed, \(\exists\) wins.)