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The Surreal Numbers and Combinatorial Games

2024/08/08 by Holden, Daniel
Computer Science · #Artificial Intelligence in Games

paper · doi:10.24382/9cg9-fb03

openalex publication_date 2024/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/15

Abstract

In the first half of this paper we study John H. Conway’s construction of the Surreal Numbers, showing it is a proper class that forms the totally ordered Field No that extends the real and ordinal numbers, and then explore some of these novel numbers,
\nsuch as w - 1, where w is the first von Neumann ordinal. In the second half we then introduce the notion of Games as a precise expression of two player perfect information sequential games, and analyse several of these Games such as Nim, Brussel Sprouts,
\nand the original Game of Borages.

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