Foundations of Mathematics [SMATM205]

This course is an introduction to proof theory via classical first-order deductive formal systems, associated semantics (model theory) and formal set theory. The first aim is to make explicit the grammatical rules, abbreviations and abuses of language that mark contemporary mathematical texts. The second is to highlight the philosophical presuppositions that govern the study of formal mathematics via the theory/metatheory dichotomy. The last is to offer powerful tools derived from set theory (e.g. ordinals) to accurately describe the hierarchies structuring contemporary mathematics.

The course is organized into three chapters. The first describes classical propositional logic (also known as Boolean logic), that of "true" and "false", interpretable by truth tables. The second focuses on first-order theories, extensions of classical logic, where we address the question of quantifiers and associated grammar rules. The example of Peano arithmetic is treated centrally, and Gödel's incompleteness theorems are explained. Finally, the last chapter develops a formal set theory. All the traditional concepts (union, parts, relation, function, Cartesian product, etc.) are reviewed, and the theory of ordinals and cardinals is studied in detail. The axiom of choice and the axiom of regularity are also studied.