Intensionality in Metamathematics and Gödel's Second Incompleteness Theorem

In metamathematics, logicians prove theorems about numbers and interpret them as being about mathematical theories. How can such an interpretation be justified? This project investigates this question and, more broadly, the phenomenon of intensionality in metamathematics, which arises from the arithmetization of mathematics. This versatile technique allows informal claims to be formalized with numbers. However, it has to be done in the right way, i.e. the arithmetization has to be faithful to the informal claims. Some mathematical research has shed new light on this subject, but much remains to be done, and an in-depth philosophical analysis is still lacking.

Through a careful logical and philosophical analysis of cutting-edge research in logic and mathematics, this project aims first to evaluate the various sources of intensionality and provide new constructions, criteria, and results in the arithmetization of the syntax and metamathematics. Second, it aims to better understand consistency statements, which will lead to a justification for the philosophical interpretations of metamathematical theorems, particularly Gödel's second incompleteness theorem, as well as a criticism of the revival of Hilbert's program.