The mathematical limits of the axiom of choice and of the axiom of determinacy

• Grant number: 19/26495-2
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: April 01, 2020
• End date: December 31, 2020
• Field of knowledge: Humanities - Philosophy - Logic
• Principal Investigator: Giorgio Venturi
• Grantee: Mahan Vaz Silva
• Host Institution: Instituto de Filosofia e Ciências Humanas (IFCH). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil

Abstract

The proof of independence of the Axiom of Choice (abbreviated AC) relative to ZF, developed by Paul Cohen in 1963, opened up new possibilities of approach to set theory. Results in descriptive set theory, the forcing method and the study of large cardinals made possible the development of set theory beyond what was imagined hitherto. One interesting axiom that may be added to ZF is the Axiom of Determinacy, which, despite being incompatible with AC, shows many results in mathematics. The objective of this project is to study large cardinals, forcing and descriptive set theory, using as line of sight Schindler's book (2014), in order to solidify the knowledge bases to a forthcoming more advanced research in set theory. Moreover, it is intended to procure lecture notes at the end of the research, material that may help researchers interested in set theory.