Entry requirements

60 credits in mathematics.

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

Assessment

Written examination at the end of the course combined with assignments given during the course.