An introduction to Conway’s games and numbers

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Schleicher, Dierk ; Stoll, Michael:
An introduction to Conway’s games and numbers.
In: Moscow Mathematical Journal. Bd. 6 (2006) Heft 2 . - S. 359-388.
ISSN 1609-3321

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Abstract

This note attempts to furnish John H. Conway's combinatorial game theory with an introduction that is easily accessible and yet mathematically precise and self-contained and which provides complete statements and proofs for some of the folklore in the subject.
Conway's theory is a fascinating and rich theory based on a simple and intuitive recursive definition of games, which yields a very rich algebraic structure. Games form an abelian GROUP in a very natural way. A certain subgroup of games, called numbers, is a FIELD that contains both the real numbers and the ordinal numbers. Conway's theory is deeply satisfying from a theoretical point of view, and at the same time it has useful applications to specific games such as Go.