The project deals with the interplay of language, logic, and mathematics. The main theme of the project is the study of the expressivity of our language when applied to mathematics concepts. The chosen perspective is that of the notion of arbitrariness, specifically in the context of set theory. The goal of this project is to investigate how modern set theory, by means of the notion of genericity can formalize and capture the notion of arbitrariness. The project is divided in three main sections: historical, philosophical, and logical.

The historical part deals with the origin of the notion of arbitrariness in the scientific, mathematical environment, of the XX century. We propose to use tools from the history of mathematics and the philosophy of language to analyze the expressive power of language. Specifically we propose to apply speech act theory to the study of mathematical language.

The philosophical part deals with the concept of set and the notion of arbitrariness. The main goal is to analyze to what extent the modern concept of set, we inherited from the axiomatic period, is capable of expressing and capturing the notion of arbitrariness. We intend to connect the notion of genericity and arbitrariness in discussing the important problem of justification of new axioms in set theory.

The logical part of the project intends to apply tools from modal logic and mathematical logic to the study of a generic perspective in set theory. This will done on the one hand developing an extensive study of the Reflexive Insensitive modal logic (RI-logic) and then applying these results in the study of the logic of generic absoluteness. On the other hand we will apply tools from model theory and set theory to the study of the models of ZFC obtained by Cohen’s method of forcing.

This project intends to coordinate the study of set theory and the foundations of mathematics between USP, UnB, UFABC, and Unicamp, together with the respective international partners.

This project is founded by FAPESP (Jovem Pesquisador grant n. 2016/25891-3) and is based at the Institute of Philosophy and Human Sciencies (IFCH) of the State University of Campinas (Unicamp).