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
- Combinatorial Set Theory / Halbeisen
- Notes on set theory / Moschovakis
- Logique et théorie des ensembles / Dehorny
- Set theory / Kunen
Notes/Handbook
Lecture notes on Moodle (423 pages).
Moodle Link
In the programs
- Semester: Spring
- Exam form: Written (summer session)
- Subject examined: Set theory
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Written (summer session)
- Subject examined: Set theory
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Written (summer session)
- Subject examined: Set theory
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional