Isaacson ’ s thesis and Wilkie ’ s theorem MSc

In this thesis, I explore Isaacson’s thesis and Wilkie’s theorem, providing philosophical and formal results on how they relate to each other. At a first approximation, Isaacson’s thesis claims that Peano arithmetic is sound and complete with respect to genuinely arithmetical statements. Using internalist notions familiar from recent work on internal categoricity theorems, I provide a formal definition of genuinely arithmetical statements. As for Wilkie’s theorem, it roughly says that, from an external perspective, Peano arithmetic is minimal, in that it is entailed by all categorical axiomatisations of the natural numbers satisfying a certain syntactic restriction. After expositing Wilkie’s theorem and the relation of its proof to other known techniques, I discuss its relation to Isaacson’s thesis, in particular whether Peano arithmetic is a maximal theory obtained from the categorical characterisation.