# Set theory

## 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 technique (obtain the reals as a countable union of countable sets, or the reals as not well-orderable, every ultirafilter on the integers is trivial). ZF with atoms and permutation models. Simulating 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 (or any equivalent course).

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 definitely lost sooner or later.

## Important concepts to start the course

- 1st order logic
- ordinal and cardinal arithmetics
- elements of proof theory
- very basic knowledge of model theory
- the compactness theorem
- Löwenheim-Skolem theorem
- the completeness theorem for 1st orderl ogic

## 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
- Formalize a model in which the reals are a countable union of countable sets
- Produce a model in which a countable set of pairs has no choice function
- Create a model in which the finite subsets of an infinite set is mapped onto the set of all its subsets

## 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 | No |

Assistants | Yes |

Forum | Yes |

## Resources

## Virtual desktop infrastructure (VDI)

No

## Bibliography

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

## Ressources en bibliothèque

- Introduction to Set theory / Hrbacek
- Set theory / Jech
- Theorie des ensembles / Krivine
- Set theory / Kunen
- Notes on set theory / Moschovakis
- Logique et théorie des ensembles / Dehorny
- Combinatorial Set Theory / Halbeisen

## Notes/Handbook

Lecture notes on Moodle (423 pages).

## Moodle Link

## In the programs

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Set theory**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Set theory**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Set theory**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks