# Coursebooks

## Set theory

Duparc Jacques

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

• 1st order logic
• basics of proof theory
• Basics of model theory
• Compacity theorem
• Löwenheim-Skolem
• Completeness theorem

#### Learning Outcomes

By the end of the course, the student must be able to:
• Specify a model of ZFC
• Prove consistency results
• Develop a generic extension
• Argue by transfinite induction
• Decide whether ZFC proves its own consistency
• Formalize the axioms of ZF, AC, CH, DC
• Sketch an inner model
• Justify the axiom of foundation

#### Teaching methods

Ex cathedra lecture and exercises

#### Expected student activities

• Attendance at lectures
• Solve the exercises

#### Assessment methods

• Writen exam (3 hours)
• Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés

#### Supervision

 Office hours Yes Assistants Yes Forum Yes

#### Resources

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

##### Notes/Handbook

Lecture notes (350 pages).

### Reference week

MoTuWeThFr
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
Under construction

Lecture
Exercise, TP
Project, other

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• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German