Fiches de cours

Set theory

MATH-318

Enseignant(s) :

Duparc Jacques

Langue:

English

Remark

Cours donné en alternance tous les deux ans

Summary

Set Theory as a foundational system for mathematics. ZF, ZFC and ZF with atoms. Relative consistency of the Axiom of Choice, the Continuum Hypothesis, the reals as a countable union of countable sets, the existence of a countable family of pairs without any choice function.

Content

Set Theory: ZFC. Extensionality and comprehension. Relations, functions, and well-ordering. Ordinals. Class and transfinite recursion. Cardinals. Well-founded relations, axiom of foundation, induction, and von Neumann's hierarchy. Relativization, absoluteness, reflection theorems. Gödel's constructible universe L. Axiom of Choice (AC), and Continuum Hypothesis inside L. Po-sets, filters and generic extensions. Forcing. ZFC in generic extensions. Cohen Forcing. Independence of the Continuum Hypothesis. HOD and AC: independence of AC. The reals without AC. Symmetric submodels of generic extensions. Applications of the symmetric submodel techinique (the reals as a countable union of countable sets, the reals not well-orderable, every ultirafilter on the integers is trivial). ZF with atoms and permutation models. Simultating permutation models by symmetric submodels of generic extensions.

Keywords

Set Theory, Relative consistency, ZFC, Ordinals, Cardinals, Transfinite recursion, Relativization, Absoluteness, Constructible universe, L, Axiom of Choice, Continuum hypothesis, Forcing, Generic extensions

Learning Prerequisites

Required courses

MATH-381 Mathematical Logic.

In particular ordinal numbers and ordinal arithmetic will be considered known and admitted.

Recommended courses

Mathematical logic (or any equivalent course on first order logic). Warning: without a good understanding of first order logic, students tend to get lost sooner orl later.

Important concepts to start the course

Learning Outcomes

By the end of the course, the student must be able to:

Teaching methods

Ex cathedra lecture and exercises

Expected student activities

Assessment methods

Supervision

Office hours Yes
Assistants Yes
Forum Yes

Resources

Virtual desktop infrastructure (VDI)

No

Bibliography

  1. Kenneth Kunen: Set theory, Springer, 1983
  2. Lorenz Halbeisen: Combinatorial Set Theory, Springer 2018
  3. Thomas Jech: Set theory, Springer 2006
  4. Jean-Louis Krivine: Theorie des ensembles, 2007
  5. Patrick Dehornoy: Logique et théorie des ensembles; Notes de cours, FIMFA ENS: http://www.math.unicaen.fr/~dehornoy/surveys.html
  6. Yiannis Moschovakis: Notes on set theory, Springer 2006
  7. Karel Hrbacek and Thomas Jech: Introduction to Set theory, (3d edition), 1999

Ressources en bibliothèque
Notes/Handbook

Lecture notes (350 pages).

Websites
Moodle Link

Dans les plans d'études

Semaine de référence

 LuMaMeJeVe
8-9     
9-10     
10-11     
11-12     
12-13     
13-14     
14-15     
15-16     
16-17     
17-18     
18-19     
19-20     
20-21     
21-22     
En construction
 
      Cours
      Exercice, TP
      Projet, autre

légende

  • Semestre d'automne
  • Session d'hiver
  • Semestre de printemps
  • Session d'été
  • Cours en français
  • Cours en anglais
  • Cours en allemand