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.
The exponential-logarithmic equivalence classes of surreal numbers
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abstract
In his monograph, H. Gonshor showed that Conway's real closed field of surreal numbers carries an exponential and logarithmic map. Subsequently, L. van den Dries and P. Ehrlich showed that it is a model of the elementary theory of the field of real numbers with the exponential function. In this paper, we give a complete description of the exponential equivalence classes in the spirit of the classical Archimedean and multiplicative equivalence classes. This description is made in terms of a recursive formula as well as a sign sequence formula for the family of representatives of minimal length of these exponential classes.
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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.