To provide the students with a basic knowledge of axiomatic and combinatorial Set Theory, both in preparation of further study of the subject and to provide tools that are useful in disciplines such as General Topology, Algebra and Functional Analysis.

The course will start with an introduction to axiomatic Set Theory, based on the axioms of Zermelo and Fraenkel.
It will show how the generally well-known facts from naive Set Theory follows from these axioms and how modern mathematics can be embedded in Set Theory.

The second part of the course will offer combinatorial tools from Set Theory that have proved useful in infinitary situations in Algebra, Topology and Analysis. These will be chosen from:

  • Partition Calculus: the theorems of Ramsey, Erdos-Rado and others
  • Combinatorial properties of families of subsets of the natural numbers
  • Trees, stationary sets, the cub filter
  • PCF theory
  • Large cardinals

Prerequisites
The course is a combination of an introductory and an advanced course in set theory. The first part provides an introduction to axiomatic set theory so that the course can be followed by a student who has no background in axiomatic set theory. We will however assume mathematical maturity, including naive use of sets that is very common in mathematics.