Institut de Mathématiques de Toulouse

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CIMI - Tinko Tinchev "Set theory as a tool "

par Delphine Dallariva - publié le , mis à jour le

Tinko Tinchev "Set theory as a tool "
Advanced course, 5 sessions in May and June 2013

Session 1 : Mercredi 22 mai, 10:00-12:00, IRIT, salle des thèses
Session 2 : Mercredi 29 mai, 10:00-12:00, IRIT, salle 001
Session 3 : Vendredi 7 juin, 10:00-12:00, IRIT, salle des thèses
Session 4 : Jeudi 13 juin, 10:00-12:00, IRIT, salle 001
Session 5 : Jeudi 20 juin, 10:00-12:00, IRIT, salle des thèses

Abstract : Set theory became popular because it constitutes a unifying system for mathematics with its own basic notions, its fundamental results and its deep open problems. As well, it has significant applications to other mathematical theories. More or less, one can say that set theory is the official language of mathematics, just as mathematics is the official language of science. The introductory university courses on set theory take often this as main goal and demonstrate a faithful representation of mathematical objects by structured sets, i.e. they consider set theory as a foundation of mathematics. The advanced courses are usually devoted to deep problems of axiomatic set theory as independence and consistency. The aim of the present course is to stress some notions and techniques that every mathematician and theoretical computer scientist needs to know : well-orderings, ordinal numbers, transfinite induction and recursion, well-founded relations, trees, (least) fixed points, Hamel basis. The following topics will be discussed in different depth and details :

Axioms of ZF
Well-ordered sets, comparability
Ordinal numbers, transfinite induction and transfinite recursion
Operations with ordinals
Goodstein sequences, Hydra game
Rank function for well-founded relation
Axiom of choice and some equivalents
Koenig’s lemma, Aronszajn’s trees, applications
(Least) Fixed points
Hamel basis, applications