Objectifs :
Présenter de façon "self-contained" la théorie des parties entières de corps réels clos et l'arithmétique de Peano. Présenter les liens avec la théorie des valuations et la géométrie réelle.

Programme :
In this mini-course, we will first systematically introduce the class of real closed fields, the celebrated Tarski transfer principle, and the valuation theory related to it (3 hours).
We will then turn to PA (Peano’s arithmetic) and the number theoretic aspects related to it. We will recall Gödel’s famous incompleteness theorem which shows that Peano arithmetic, and indeed any recursive fragment of number theory extending it, is not complete. This theorem produced great interest in studying the deductive power of recursive fragments of number theory. Next we will turn to integer parts of real closed fields, and discuss Shepherdson’s theorem cited above. We will then proceed to show that every real closed field admits an integer part (which is a model of the fragment OI (Open Induction) of PA), a result obtained by a modification of I. Kaplansky’s embedding for real closed fields into fields of generalized power series (3 hours).
We will discuss the arithmetic properties of those rings, such as their irreducible elements, and address questions such as the existence of co-final sequences of prime elements in these rings (3 hours).
In the last part of the course we will turn our attention to real closed fields which admit an integer part whose non-negative cone is a model of PA. We call such a field an IPA − RCF. We investigate the valuation theoretical properties of any IPA − RCF. We show that a non-archimedean IPA − RCF admits exponentiation. We close by a discussion of real closed exponential fields (3 hours).

Pré-requis :
Niveau L3 en géométrie réelle et en logique, M1 en algèbre commutative et géométrie algébrique

Compétences acquises à l'issue de la formation :
Logique du 1er ordre (théorème de Gödel etc) concernant les fondements de l'arithmétique et en lien avec la géométrie réelle (théorie des valuations, o-minimalité, exponentiation).