The Surreal Numbers and Combinatorial Games

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, such 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, and the original Game of Borages.