Set theory and its interal logic

Abstract

With this project I propose to investigate the phenomenon of independence in set theory in the context of paraconsistent logic, by a multidisciplinary point of view. The main motivation of this work comes from the limits of classical logic in the study of the notion of truth in the models of ZFC (i.e the standard first order axiomatization of set theory). In particular I plan to apply a Logic of Formal Inconsistency (LFI) in the study of the mathematical structure that consists of all possible models that we can obtain by means of the method of forcing (the multiverse). The first goal is to determine which are the best paraconistent logical principles that can describe the multiverse, while the second - and main - goal is to determine if and how it is possible to perform independence proofs in a paraconsitent set theory (i.e. a set theory whose internal logic is paraconsitent; in particular an LFI). In particular I plan to define and to study a paraconsistent version of the constructible class $L$ and a paraconsistent version of the method of forcing. Besides, my research project aims to give a philosophical justification of the use of a paraconsistent logic in the study of independence in set theory, discussing its quasi-empirical character and the consequences that this aspect has on the notion of truth in set theory.