# An introduction to the theory of infinite games, with examples from infinite chess, University of Connecticut, December 2014

This will be a talk for the interdisciplinary Group in Philosophical and Mathematical Logic at the University of Connecticut in Storrs, on December 5, 2014.

Abstract. I shall give a general introduction to the theory of infinite games, with a focus on the theory of transfinite ordinal game values. These ordinal game values can be used to show that every open game — a game that, when won for a particular player, is won after finitely many moves — has a winning strategy for one of the players. By means of various example games, I hope to convey the extremely concrete game-theoretic meaning of these game values for various particular small infinite ordinals. Some of the examples will be drawn from infinite chess, which is chess played on a chessboard stretching infinitely without boundary in every direction, and the talk will include animations of infinite chess positions having large numbers of pieces (or infinitely many) with hundreds of pieces making coordinated attacks on the chessboard. Meanwhile, the exact value of the omega one of chess, denoted $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$, is not currently known.

## 2 thoughts on “An introduction to the theory of infinite games, with examples from infinite chess, University of Connecticut, December 2014”

1. Hi Professor Hamkins! You might remember me from two of your courses at Berkeley. I wrote a paper on weak tautologies, around fuzzy math, in your intro to mathematical logic course (and got my grade bumped to a B+ as a result, if I remember correctly). I remember telling you at my graduation that your award as outstanding lecturer was well-deserved, and I meant it! I don’t know much about the decidability of the mate-in-n problem of infinite chess, but I do like to play from time to time 😉 If you might like to get together, send me an email, and take care! Steve.

• Thanks for the kind words, Steve, both then and now. I do remember you and your paper—it was about considering various senses of tautology in fuzzy logic, where the truth value is always 1, or always more than 1/2, or always at least 1/2, and so on. I still find that interesting… -JDH