Modelos da teoria de conjuntos não clássicas

Author(s)

Daniel Santiago Jockwich Martinez

• Document type: Doctoral Thesis
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Filosofia e Ciências Humanas
• Defense date: 2021-12-15

Examining board members

Giorgio Venturi; Graham Priest; Sourav Tarafder; Rodrigo de Alvarenga Freire; Ekaterina Kubyshkina

• Advisor: Rodolfo Cristian Ertola Biraben; Giorgio Venturi

Abstract

The essential weakness of non-classical set theories is their lack of natural models. In particular, we lack models that are mathematically expressive. In this thesis, we aim to tackle this problem for several classes of non-classical set theories. We provide models of paraconsistent and paracomplete set theories in the form of algebra-valued models. More especifically, we construct a class of paraconsistent models of the negation-free fragment of ZF and we build a class of non-classical models of ZF which are neither paraconsistent nor paracomplete. Then, we explore two different extensions of this work: (1) expanding the language of the underlying algebra with different operators and (2) modifying the interpretation of set membership and equality in our algebra-valued models. This gives rise to a several classes of paraconsistent models of set theory and to a class paracomplete models of set theory. Moreover, we show that these models are different from each other and that we can construct paraconsistent model of ZFC based on Priest's Logic of Paradox. We believe that this suggests that non-classical set theories and, in particular, paraconsistent set theories can capture a reasonable amount of standard mathematics (AU)

• FAPESP's process: 17/23853-0 - Arbitrariness and definability in the context of non-classical logics
• Grantee: Daniel Santiago Jockwich Martinez
• Support Opportunities: Scholarships in Brazil - Doctorate (Direct)