# The omega one of chess

## Joel David Hamkins

### The City University of New York

This talk will be based on my recent paper with C. D. A. Evans, Transfinite game values in infinite chess.

Infinite chess is chess played on an infinite chessboard. Since checkmate, when it occurs, does so after finitely many moves, this is technically what is known as an *open* game, and is therefore subject to the theory of open games, including the theory of ordinal game values. In this talk, I will give a general introduction to the theory of ordinal game values for ordinal games, before diving into several examples illustrating high transfinite game values in infinite chess. The supremum of these values is the *omega one of chess*, denoted by $omega_1^{mathfrak{Ch}}$ in the context of finite positions and by $omega_1^{mathfrak{Ch}_{hskip-2ex atopsim}}$ in the context of all positions, including those with infinitely many pieces. For lower bounds, we have specific positions with transfinite game values of $omega$, $omega^2$, $omega^2cdot k$ and $omega^3$. By embedding trees into chess, we show that there is a computable infinite chess position that is a win for white if the players are required to play according to a deterministic computable strategy, but which is a draw without that restriction. Finally, we prove that every countable ordinal arises as the game value of a position in infinite three-dimensional chess, and consequently the omega one of infinite three-dimensional chess is as large as it can be, namely, true $omega_1$.

Professor Hamkins (Ph.D. 1994 UC Berkeley) conducts research in mathematical and philosophical logic, particularly set theory, with a focus on the mathematics and philosophy of the infinite. He has been particularly interested in the interaction of forcing and large cardinals, two central themes of contemporary set-theoretic research. He has worked in the theory of infinitary computability, introducing (with A. Lewis and J. Kidder) the theory of infinite time Turing machines, as well as in the theory of infinitary utilitarianism and, more recently, infinite chess. His work on the automorphism tower problem lies at the intersection of group theory and set theory. Recently, he has been preoccupied with various mathematical and philosophical issues surrounding the set-theoretic multiverse, engaging with the emerging debate on pluralism in the philosophy of set theory, as well as the mathematical questions to which they lead, such as in his work on the modal logic of forcing and set-theoretic geology.