Welcome to Foundations of Logic and Mathematics! The course is divided in two parts: the first half will be taught by Matteo Plebani, and the other half will be led by Lorenzo Rossi.
In the first part, we will analyze the question of how we learn mathematical language, with a focus on the language of arithmetic. We will review classical limitative results (Löwenheim-Skolem theorems and the compactness theorem, the halting problem, Gödel’s first incompleteness theorem, and Tarski’s undefinability theorem). A discussion of the philosophical significance of these results will follow. We will also discuss whether computational structuralism provides a satisfactory answer to the question of what fixes the reference of our arithmetical vocabulary.
The second part will deal with second-order solutions to the problem of referential indeterminacy in mathematics. After an introduction to second-order logic (SOL), with full and Henkin semantics, we will show that SOL does not share some of the meta-logical features of first-order logics (in particular, compactness and the Löwenheim-Skolem theorems), and can therefore single out unique models (up to isomorphism) of fundamental mathematical theories. More specifically, we will introduce the phenomenon of categoricity, and show that the second-order theories of the main number systems are categorical. If time permits, we will show that a quasi-categoricity result holds for second-order theories of sets.
In the first part, we will analyze the question of how we learn mathematical language, with a focus on the language of arithmetic. We will review classical limitative results (Löwenheim-Skolem theorems and the compactness theorem, the halting problem, Gödel’s first incompleteness theorem, and Tarski’s undefinability theorem). A discussion of the philosophical significance of these results will follow. We will also discuss whether computational structuralism provides a satisfactory answer to the question of what fixes the reference of our arithmetical vocabulary.
The second part will deal with second-order solutions to the problem of referential indeterminacy in mathematics. After an introduction to second-order logic (SOL), with full and Henkin semantics, we will show that SOL does not share some of the meta-logical features of first-order logics (in particular, compactness and the Löwenheim-Skolem theorems), and can therefore single out unique models (up to isomorphism) of fundamental mathematical theories. More specifically, we will introduce the phenomenon of categoricity, and show that the second-order theories of the main number systems are categorical. If time permits, we will show that a quasi-categoricity result holds for second-order theories of sets.
- Teacher: Matteo Plebani
- Teacher: Lorenzo Rossi