Unifying universes: a new take on NF

• Grant number: 22/06486-1
• Support Opportunities: Generation Project Research Grant
• Start date: September 01, 2023
• End date: August 31, 2028
• Field of knowledge: Humanities - Philosophy - Logic
• Principal Investigator: Daniel Santiago Jockwich Martinez
• Grantee: Daniel Santiago Jockwich Martinez
• Host Institution: Instituto de Filosofia e Ciências Humanas (IFCH). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated researchers: Bruno Ramos Mendonça ; Giorgio Venturi ; Henrique Antunes Almeida ; Rodrigo de Alvarenga Freire
• Associated scholarship(s): 24/03120-1 - Algebra-valued models of type theory, BP.GR

Abstract

This project aims to bring together the study of classicality and non-classicality within the fields of mathematical logic and philosophy of logic. Our overarching hypothesis is that real progress can only be made by means of a new synthesis, which will tackle questions related to both fields in a unified way. The project is divided in three parts. The first part is the technical core of the project. We will unify classical and non-classical foundations of mathematics within a set-theoretic framework. In particular, we plan to show that any non-classical logic which satisfies some minimal constraints can give rise to a model of ZFC and that these models are mathematically expressive. The second part takes a philosophical perspective. We will try to understand the similarities and differences between set-theoretic and logical pluralism. We hope that might give us new insights in both areas. The final part of this project concerns applications. We will tackle a notorious open conjecture in the area of mathematical logic: the consistency of Quine´s New Foundation. In particular, we plan to apply the model constructions that we will develop in the first part of the project to provide a model for New Foundation. (AU)