# Coursebooks

## Set theory

Duparc Jacques

English

#### Remarque

Cours donné en alternance tous les deux ans (donné en 2019-20)

#### Summary

Set Theory as a foundational system for mathematics. Relative consistency of the Axiom of Choice and the Continuum Hypothesis.

#### 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, 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 the Axiom of Choice: independence of the Axiom of Choice.

#### 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

##### Bibliography

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

### In the programs

• Mathematics - master program, 2019-2020, Master semester 2
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Set theory
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Applied Mathematics, 2019-2020, Master semester 2
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Set theory
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Applied Mathematics, 2019-2020, Master semester 4
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Set theory
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Communication Systems - master program, 2019-2020, Master semester 2
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Set theory
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Communication Systems - master program, 2019-2020, Master semester 4
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Set theory
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Computer Science - Cybersecurity, 2019-2020, Master semester 2
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Set theory
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Computer Science - Cybersecurity, 2019-2020, Master semester 4
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Set theory
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Computer Science, 2019-2020, Master semester 2
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Set theory
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Computer Science, 2019-2020, Master semester 4
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Set theory
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks

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10-11 CM012
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Lecture
Exercise, TP
Project, other

### legend

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