This will be a talk for the Mathematics Graduate Student Seminar on 23rd October 2026, 5pm at the CUNY Graduate Center in midtown Manhattan.
On the fundamental theorem of finite games
Abstract: We will give five different proofs of the fundamental theorem of finite games, due originally to Zermelo (1913), showing that in any two-player game of perfect information, one of the players has a winning strategy or both players have drawing strategies. We will discuss how the proofs generalize to infinite games, giving several different proofs of the Gale-Stewart theorem on open determinacy, and also establish the existence of nondetermined infinite games.
The talk will be based in part of my “Short Stories” essay that appeared in the AMS Notices:
- Joel David Hamkins. The fundamental theorem of finite games. Notices AMS 72.8 (2025). Short Stories column, pp. 858-860. doi:10.1090/noti3203.
See also my substack essay:
- Joel David Hamkins, The fundamental theorem of finite games, Infinitely More, March 2025. https://www.infinitelymore.xyz/p/fundamental-theorem-of-finite.
In the talk, I will go beyond both these essays, to speak of game indeterminacy and the proof from the axiom of choice that there are nondetermined games.