Surreal numbers can be presented as sets of ordinals with a maximal 'birthday' element, giving a set-theoretic foundation equivalent to Gonshor's sign expansions and Conway's games.
An Introduction to Conway's Games and Numbers
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abstract
This is an introduction into John Conway's beautiful Combinatorial Game Theory, providing precise statements and detailed proofs for the fundamental parts of his theory. (1) Combinatorial game theory, (2) the GROUP of games, (3) the FIELD of numbers, (4) ordinal numbers, (5) games and numbers, (6) infinitesimal games, (7) impartial games.
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On Conway's Numbers and Games, the Von Neumann Universe, and Pure Set Theory
Surreal numbers can be presented as sets of ordinals with a maximal 'birthday' element, giving a set-theoretic foundation equivalent to Gonshor's sign expansions and Conway's games.