Set Theory
5 credits
Syllabus, Bachelor's level, 1MA031
A revised version of the syllabus is available.
- Code
- 1MA031
- Education cycle
- First cycle
- Main field(s) of study and in-depth level
- Mathematics G2F
- Grading system
- Pass with distinction (5), Pass with credit (4), Pass (3), Fail (U)
- Finalised by
- The Faculty Board of Science and Technology, 24 April 2013
- Responsible department
- Department of Mathematics
Entry requirements
Mathematics 60 credits
Learning outcomes
In order to pass the course (grade 3) the student should
- be able to formalise mathematical statements in ZF set theory;
- master cardinal and ordinal arithmetic;
- be able to apply variants of the axiom of choice;
- be able to carry out proofs and constructions by transfinite induction and recursion;
- be familiar with various paradoxes in naive set theory and understand the need for formalisation of set theory;
- know independence results for the continuum hypothesis and the axiom of choice;
- be able to give an account of the basic concepts in category theory;
- be able to present mathematical arguments to others.
Content
Paradoxes. The cumulative hierarchy. The axioms for Zermelo-Fraenkel's set theory. Classes. Ordered sets: partial and linear orderings, well-founded relations, well-orderings. The axiom of choice and equivalent variants. Zorn's lemma and the well-ordering principle. Transfinite induction and recursion. Ordinals and cardinals. The continuum hypothesis. Briefly about independence results and models of set theories. Briefly about category theory.
Instruction
Lectures and problem solving sessions.
Assessment
Written examination at the end of the course combined with assignments given during the course.