6 ECTS credits
150 h study time
Offer 1 with catalog number 4013341ENR for all students in the 2nd semester at a (E) Master - advanced level.
The classical paradoxes, such as Russell's paradox and the Burali-Forti paradox in the foundations of mathematics, motivate an axiomatic description of set theory. Another motivation for the introduction of formal set theory is that the solution of several open problems in abstract analysis, measure theory and topology have been shown to depend on basic axioms for sets such as the continuum hypothesis, Martin's axiom or the Souslin hypothesis.
The course starts from naive set theory. In this context the theory of well-ordered sets, ordinals and cardinals is developed. After a brief revision of formal logic, the Zermelo Fraenkel (ZF) axioms are introduced. The axiom of choice and its most important implications in mathematics are discussed.
Consistency results are treated. The most famous example of a statement independent of ZFC is the continuum hypothesis but also other independent statements, coming from various branches of mathematics are discussed.
A foundation of category theory is presented: the existence of a universe is discussed and its relation with the axiom on the existence of a strongly inaccessible cardinal is treated.
1. Naive set theory
- Well ordering
- Ordinals
- Paradoxes of Russell and Burali-Forti
2. Axiomatic set theory
- The language of set theory
- Zermelo Fraenkel axioms
- Classes
- Recursion
3. The axiom of choice
- Equivalent formulations
- Applications in mathematics
4. Cardinals
- Equipotent sets
- Representation of infinite cardinals
- Continuum hypothesis
5. Consistency
- Formal deduction
- Relative consistency
- Examples of independent statements
6. Well founded sets
- Axiom of foundation
- Constructible universe
7. Existence of a universe
-strongly inaccessible cardinals
8. Incompleteness theorem of Gödel
Course notes are available
- The student has insight in the importance of an axiomatic foundation of mathematics and in the impact it has on various fields, such as measure theory, analysis, topology and algebra.
- The student knows different formulations of the axiom of choice and the impact it has outside set theory
- The student can perform calculations with infinite cardinals and can apply this ability.
- The student understands the difference between regular and singular cardinals, knows the meaning of the axiom stating that strongly inaccessible cardinals exist and knows the relation with the existence of universes.
- The student knows the meaning of relative consistency and knows examples of undecidable formulas in ZFC.
- The student has an overall insight in the material, has a deep understanding of new concepts and results and is aware of the connection between de various concepts.
- The student has insight in the relation to analogous concepts as they were introduced in other courses.
- The student can analyse proofs and understands the logical reasoning behind them. For a given proposition the student is aware of the role the conditions play in the proof.
- The student can complete proofs that are left as an exercise or that are only partially explained in the syllabus or in class. The missing arguments can be filled in independently.
- The student masters the mathematical language and is able to produce correct mathematical formulations and proofs.
- The student can independently solve problems: he/she is able to recognize a problem, to choose an appropriate strategy, to select the most suitable method
The final grade is composed based on the following categories:
Oral Exam determines 100% of the final mark.
Within the Oral Exam category, the following assignments need to be completed:
This offer is part of the following study plans:
Master of Mathematics: Fundamental Mathematics (only offered in Dutch)
Master of Mathematics: Education (only offered in Dutch)
Master of Teaching in Science and Technology: wiskunde (120 ECTS, Etterbeek) (only offered in Dutch)