Zermelo's theorem can be applied to all finite-stage two-player games with complete information and alternating moves.
The first one covered axiomatic and genetic methods in the foundation of mathematical disciplines, and the second speech was on the game of chess.
Zermelo's original paper describing the theorem, Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels, was published in German in 1913.
[1] Zermelo considers the class of two-person games without chance, where players have strictly opposing interests and where only a finite number of positions are possible.
Although in the game only finitely many positions are possible, Zermelo allows infinite sequences of moves since he does not consider stopping rules.
Then he addresses two problems: To answer the first question, Zermelo states that a necessary and sufficient condition is the nonemptyness of a certain set, containing all possible sequences of moves such that a player wins independently of how the other player plays.
This set may also be empty, i.e., the player can avoid his loss for only finitely many moves if his opponent plays correctly.
In 1927, a Hungarian mathematician Dénes Kőnig revised Zermelo's paper and pointed out some gaps in the original work.
However, this argument is not correct because Zermelo considered two positions different whether Black or White makes a move.
However, recent research on the Zermelo's theorem demonstrates that backward induction was not used to explain the strategy behind chess.
This method analyses the game starting at the end, and then works backwards to reach the beginning.
Kalmár generalised the work of Zermelo and Kőnig in his paper "On the Theory of Abstract Games".