Learning outcomes

Good mathematical notions are required, as the course is connected to fields like logic, analysis, geometry, ...

Goals

 

This course aims to give the bases of the Mathematical logic and the Set Theory. It shows how one can indeed build an axiomatic formal theory that can produce all the mathematics and, for this reason, it can give to the mathematician students the satisfaction to have gone once till the source of their knowledge.

Content

The course has three axes. The first axis is more philosophical. It questions the concept of truth in sciences and in mathematics in particular. How the mathematicians see the relation between the truth and the reality; what does it mean when one says : this proposal is proven or refuted. ¿¿ The second axis is logical. It makes the distinction between the syntactic level and the interpretative level (semantic) of a proposition. Theories are built and theorems are proven. The logic of the first order is studied, amongst other things, in its properties of coherence and completeness. The third axis is axiomatic. We try here to build the set theory starting from Zermelo-Fraenkel fundamental axioms. The construction of ordinal and cardinal numbers is presented in detail. The axiom of the choice and the axiom of foundation are studied. Finally, the Godel theorems of incompleteness are explained.

Table of contents

Roots: the logic of Aristotle The vocabulary and formalism: language, model, structure, coherence, completness The logic of the first order predicates. The set theory and the construction of numbers The question of the infinity : the cardinal classes and the aleph numbers The question of auto-referential models and the Gödel theorems Conclusions

Exercices

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Assessment method

The examination is done orally. Each student prepares a subject about one of the two most important parts of the course (logical of the predicates or set theory), which he presents during 10 minutes, then he is questioned on another part of the course (with 10 minutes preparation).

Sources, references and any support material

J.-L. KRIVINE, Théorie axiomatique des ensembles, P.U.F., 2e éd. 1972

Language of instruction

Français
Training Study programme Block Credits Mandatory
Standard 0 3
Standard 0 3
Standard 0 3
Standard 0 3
Standard 0 3
Standard 1 3
Standard 2 3
Standard 2 3
Standard 2 3
Standard 2 3