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, 15 March 2007
Responsible department: Department of Mathematics

Entry requirements

Mathematics 60 credit points

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. The continuum hypothesis. Briefly about independence results and models of set theories. Briefly about category theory.

Instruction

Lectures and problem solving sessions.

Assessment

Written and, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course.