On Conway's Numbers and Games, the Von Neumann Universe, and Pure Set Theory
Wolfgang Bertram
TL;DR
The paper embeds Conway's surreal numbers within the pure set-theoretic universe, modeling them as sets of ordinals with a birthday and constructing a full binary number tree in the von Neumann hierarchy. It develops sign-expansions and canonical cuts to define arithmetic (addition, multiplication) and order, showing equivalence with established surreal-number formalisms and linking to the nimber-field in impartial game theory. By building Conway reals and their sign-interpretations, it demonstrates a natural, algorithmic embedding of $ extbf{R}$ into the Conway hierarchy and extends these ideas to graded, partizan settings via a graded von Neumann universe, bridging CGT with pure set theory. The work argues for a unified, intrinsic view where Conway numbers, ordinals, and the von Neumann universe form a trinity, and outlines open problems and a program for intrinsic surreal-number theory, including the omega-map and surcomplex numbers, with potential implications for the foundations and reach of mathematical structure.
Abstract
We take up Dedekind's question ''Was sind und was sollen die Zahlen?'' (''What are numbers, and would should they be?''), with the aim to describe the place that Conway's (Surreal) Numbers and Games take, or deserve to take, in the whole of mathematics. Rather than just reviewing the work of Conway, and subsequent one by Gonshor, Alling, Ehrlich, and others, we propose a new setting which puts the theory of surreal numbers onto the firm ground of ''pure'' set theory. This approach is closely related to Gonshor's one by ''sign expansions'', but appears to be significantly simpler and clearer, and hopefully may contribute to realizing that ''surreal'' numbers are by no means surrealistic, goofy or wacky. They could, and probably should, play a central role in mathematics. We discuss the interplay between the various approaches to surreal numbers, and analyze the link with Conway's original approach via Combinatorial Game Theory (CGT). To clarify this, we propose to call pure set theory the algebraic theory of pure sets, or in other terms, of the algebraic structures of the von Neumann universe. This topic may be interesting in its own right: it puts CGT into a broad context which has a strong ''quantum flavor'', and where Conway's numbers (as well as their analogue, the nimbers) arise naturally.