V Encontro da Pós-Graduação em Matemática da UFBA 20 a 24 de novembro de 2017

An algebraic framework to a theory of sets based on the surreal numbers Hugo Luiz Mariano

∗

Abstract This is the PhD thesis theme of Dimi Rocha Rangel at IME-USP, a MSc by IMEUFBA (2012). The notion of surreal number was introduced by J.H. Conway in the mid 1970’s (see [Con]): the surreal numbers constitute a linearly ordered (proper) class N o that, working within the background set theory NBG, can be defined by a recursion on the class On of all ordinal numbers. Since then, have appeared many constructions of this class and was isolated a full axiomatization of this notion that been subject of interest due to large number of interesting properties they have, including model-theoretic ones. Such constructions suggests strong connections between the class N o of surreal numbers and the classes V and On, of all sets and all ordinal numbers. In an attempt to codify the universe of sets directly within the surreal number class, we have founded some clues that suggest that this class is not suitable for this purpose. Carefully formalizing the definition of the class of pre-numbers (and some variants), which is an intermediate stage in the construction of the Conway surreal numbers, we obtain structures which have copies of the universe of sets as well as copies of the class of surreal numbers. Thus, in particular, we gave first steps toward a certain kind of "relative set theory", in this new setting. The main aim of this work is to isolate and explore properties of these new constructions and present the notion of (partial) SUR algebra, an attempt to obtain an "algebraic theory for surreal numbers" along the lines of the Algebraic Set Theory of Joyal and Moerdijk ([JM]): to establish (abstract and general) links between the class of all surreal numbers and a universe of "surreal sets" similar to the relations obtained by Joyal and Moerdijk between the class of all ordinals and the class of all sets, that respects and expands the links between the linearly ordered class of all ordinals and of all surreal numbers.

References [Con] J.H. Conway, On numbers and games, A. K. Peters Ltd., second ed., 2001. [JM] A. Joyal, I. Moerdijk, Algebraic set theory, London Mathematical Society, L.N.S. v.220, Cambridge University Press, 1995. ∗

USP, [email protected]

1