Scope, Limitations and Extensions of Gödel's Incompleteness Theorems

• Grant number: 24/07603-7
• Support Opportunities: Scholarships abroad - Research
• Start date: December 01, 2025
• End date: February 28, 2026
• Field of knowledge: Humanities - Philosophy - Logic
• Principal Investigator: Walter Alexandre Carnielli
• Grantee: Walter Alexandre Carnielli
• Host Investigator: Marco Panza
• Host Institution: Centro de Lógica, Epistemologia e História da Ciência (CLE). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Institution abroad: Chapman University, United States
• Associated research grant: 20/16353-3 - Rationality, logic, and probability: RatioLog, AP.TEM

Abstract

The present project is centered on a broad theme that follows my research line, understood as the clarification and philosophical interpretation of paraconsistency, its applications, and implications. In particular, the objective is to analyse Gödel's Incompleteness Theorems in the context of paraconsistency, evaluating their impact and importance in the philosophy of science and the philosophy of mathematics.The distinction between the concepts of consistency and non-contradiction, contradiction and non-consistency, as well as inconsistency and non-consistency, along with the consequent distinction between contradiction and triviality, are fundamental principles of the Logics of Formal Inconsistency (LFI's), which form the basis for much of the current paraconsistent systems.The project aims to demonstrate that these principles will lead to different proposals for formalizing Gödel's theorems, affecting the proofs and the scope of these theorems. The procedures of Gödel, arithmetization, as well as the fixed-point lemma (diagonalization), will not be questioned, assumed as valid (even if changing the base logic). Unlike other authors, who argue that Peano, Russell, Hilbert, and Gödel were using the "wrong logic" to formalize arithmetic, and that another arithmetic would have better properties, here the intention is not to change the formulation of arithmetic, but rather to evaluate the essential logical assumptions that guarantee Gödel's proof, and to what extent more subtle logical systems, such as paraconsistent ones, maintain such guarantee.This project can be described as a critical reconstruction of Gödel's proofs, questioning their scope, and consequently their limitations, and potential extensions. (AU)