Set theory and its interal logic

• Grant number: 13/25095-4
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: March 01, 2014
• End date: February 13, 2017
• Field of knowledge: Humanities - Philosophy - Logic
• Principal Investigator: Walter Alexandre Carnielli
• Grantee: Giorgio Venturi
• Host Institution: Centro de Lógica, Epistemologia e História da Ciência (CLE). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 10/51038-0 - Logical consequence, reasoning and computation - LOGCONS, AP.TEM
• Associated scholarship(s): 14/25342-4 - A logical analysis of the notion of consistency in set theoretical practice, BE.EP.PD

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.

Scientific publications

(6)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

GIORGIO VENTURI. A note on the introduction of Hilbert’s Grundlagen der Geometrie. Manuscrito, v. 40, n. 2, p. 5-17, 2017-06-00. (14/25342-4, 13/25095-4)

GILBERT, DAVID R.; VENTURI, GIORGIO. Neighborhood Semantics for Logics of Unknown Truths and False Beliefs. AUSTRALASIAN JOURNAL OF LOGIC, v. 14, n. 1, SI, p. 246-267, 2017. (13/25095-4)

VENTURI, GIORGIO. A note on the introduction of Hilbert's Grundlagen der Geometrie. Manuscrito, v. 40, n. 2, p. 5-17, APR-JUN 2017. (13/25095-4, 14/25342-4)

GILBERT, DAVID R.; VENTURI, GIORGIO. A note on logics of essence and accident. LOGIC JOURNAL OF THE IGPL, v. 28, n. 5, p. 881-891, OCT 2020. (14/25342-4, 13/25095-4)

GILBERT, DAVID R.; VENTURI, GIORGIO. REFLEXIVE-INSENSITIVE MODAL LOGICS. Review of Symbolic Logic, v. 9, n. 1, p. 167-180, MAR 2016. (13/25095-4)

GILBERT, DAVID R.; VENTURI, GIORGIO. Neighborhood Semantics for Logics of Unknown Truths and False Beliefs. AUSTRALASIAN JOURNAL OF LOGIC, v. 14, n. 1, p. 22-pg., 2017-01-01. (13/25095-4)