My PhD project investigates the phenomenon of intensionality in metamathematics and, more specifically, in Gödel's second incompleteness theorem. It aims first to critically evaluate the different sources of intentionality due to the arithmetization of the syntax, and to defend a criterion of admissibility of numbering as well as general conditions under which an arithmetical formula can express a syntactic property. Second, it explores the philosophical interpretations of Gödel's second incompleteness theorem and how one should justify it.
I am also working on more general issues in philosophy of mathematics and computation such as finitism, Isaacson's thesis, deviant notation and arbitrary objects. In my master's thesis, I defended a new interpretation of the principle of generic attribution for arbitrary objects and sketched some potential applications in the philosophy of mathematics.
The choice of Gödel numberings is one of the sources of indeterminacy in metamathematics; for certain numberings, our well-known theorems, such as Gödel's second incompleteness theorem, do not hold. However, it is widely believed that all our metamathematical theorems hold for all reasonable Gödel numberings. This article investigates this notion of reasonable Gödel numberings and provides a general framework for answering philosophical questions about Gödel numberings and the arithmetisation of syntax.
Tentative titles and summaries.
In this work, we introduce a new class of numberings — uniform numberings — and investigate their properties to prove the consistency of some basic untyped truth theories with respect to the class of uniform numberings. This is joint work with Balthasar Grabmayr.
I discuss Detlefsen's stability problem and, more generally, how to argue in favour of the standard philosophical interpretation of Gödel's second incompleteness theorem.