Axiomatic foundations of class forcing and symmetric extensions.

• Grant number: 21/11766-0
• Support Opportunities: Research Grants - Visiting Researcher Grant - Brazil
• Start date: January 01, 2022
• End date: December 31, 2022
• Field of knowledge: Humanities - Philosophy - Logic
• Principal Investigator: Giorgio Venturi
• Grantee: Giorgio Venturi
• Visiting researcher: Rodrigo de Alvarenga Freire
• Visiting researcher institution: Universidade de Brasília (UnB). Instituto de Ciências Humanas (ICH), Brazil
• Host Institution: Instituto de Filosofia e Ciências Humanas (IFCH). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil

Abstract

This project aims to develop the new foundational approach to forcing which was initially proposed in "An Axiomatic Approach to Forcing and Generic Extensions", by Rodrigo Freire, covering cases that were left open. At the same time, in order to achieve this end goal, we will provide a conceptual analysis of the relevant notions as an attempt to illuminate forcing in a general setting.Forcing and generic extensions are usually not given as realizations of a concept, rather they are presented as specific constructions serving a specific purpose. Indeed, there are many different constructions with the same effect and differing on technical minutiae which obfuscate its essential components. If we want to make explicit what is this specific purpose, we must first capture the general idea avoiding inessential variations.In order to accomplish that, we turn towards an axiomatic approach. The situation is analogous to that of the real number system up to isomorphism: There are many different constructions of this system, via Dedeking cuts or Cauchy sequences, for example, but the axiomatic approach gives us a concept behind those constructions. We wish to capture an axiomatic conceptual basis for forcing and generic extensions. (AU)