Zermelo 1913

Abstract

Zermelo’s 1913, Über eine Anwendung der Mengenlehre auf die Theorie des Schachpiels, is an account of an address given at the Fifth International Congress of Mathematicians in Cambridge in 1912. It is often cited as the first mathematical analysis of strategies in games. While the paper claims to be an application of set theory, and while it would have appeared that way to Zermelo’s contemporaries, the set-theoretic notions in the paper have since become part of standard mathematical practice, and to modern eyes the arguments in the paper are more combinatorial than set-theoretic. The notion of “Zermelo’s Theorem” (usually described as a variant of “in chess, either White or Black has a winning strategy, or both can force a draw”) derives from this paper. Although statements of this sort follow from the claims made in the paper, Zermelo’s arguments for these claims are incomplete. As we shall see below, there are other gaps in the paper, one of which was fixed by Dénes König in his 1927a. König’s paper also contains two paragraphs, 1927b, on arguments of Zermelo fixing this gap, using ideas similar to König’s.