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

• Ano de defesa: 2018
• Autor(a) principal: Rangel, Dimi Rocha
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Algebraic set theory; Números surreais; SUR-algebras; SUR-álgebras; Surreal numbers; Teoria algébrica dos conjuntos
• Link de acesso: http://www.teses.usp.br/teses/disponiveis/45/45131/tde-02032019-143956/

The notion of surreal number was introduced by J.H. Conway in the mid 1970\'s: the surreal numbers constitute a linearly ordered (proper) class No containing the class of all ordinal numbers (On) that, working within the background set theory NBG, can be defined by a recursion on the class On. 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 No of surreal numbers and the classes 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. The present work is an attempt to obtain an \"algebraic (set) theory for surreal numbers\" along the lines of the Algebraic Set Theory - a categorial set theory introduced in the 1990\'s: to establish abstract and general links between the class of all surreal numbers and a universe of \"surreal sets\" similar to the relations between the class of all ordinals (On) and the class of all sets (V), that also respects and expands the links between the linearly ordered class of all ordinals and of all surreal numbers. We have introduced the notion of (partial) surreal algebra (SUR-algebra) and we explore some of its category theoretic properties, including (relatively) free SUR-algebras (SA, ST). We have established links, in both directions, between SUR-algebras and ZF-algebras (the keystone of Algebraic Set Theory). We develop the first steps of a certain kind of set theory based (or ranked) on surreal numbers, that expands the relation between V and On.