Some notions of mathematical logic [ LMAT1235 ]

We start with a naive point of view on sets. In this framework we introduce ordinals and cardinals, and we develop an elementary theory which shows very clearly why this naive point of view is untenable. 

Then we look at the axiomatic set theory of Zermelo and Fraenkel. We are especially concerned with independence and inconsistency problems, taking as examples the axiom of choice and the continuum hypothesis. 

In parallel with this we develop a base of propositional and predicate calculus, i.e. the general properties of first order structures and languages, which we need to fully understand the problems which appear in set theory.

 

 

 

We aim at making explicit the laws of mathematical reasoning, when it is presented as a formalised theory. We examine the peculiarities of the languages that are used, the propositions that are taken as starting points, the deduction rules that are usually admitted. 

As an example we consider naive set theory with its formalisation ZF.
We focus on the limitations of the formalisation process, e.g. the impossibility to guarantee a definitive rigour. 

The spirit and the presentation are of the same kind as for any mathematics course: we give definitions, we construct chains of propositions, we prove theorems.

 

Combination of a series of lectures on theoretical aspects and of a series of exercises about examples and applications.

 

Contents
1. Introduction

2. Naive set theory: ordinals and cardinals

3. Axiomatic set theory: ZF, the axiom of choice, consistency

4. Propositional and predicate logic