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 naïve Set Theory follow from these axioms and how modern mathematics can be embedded in Set Theory.
The second part of the course will be devoted to more advanced topics in Set Theory.
Possible topics are
- The Constructible Universe
- 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
The course is a combination of an introductory and an advanced course in set theory. Since we begin by developing Set Theory from its axioms the course can be taken by students without earlier experience of axiomatic set theory. We will however assume mathematical maturity, including the naïve use of sets that is very common in mathematics.
Rules about Homework / Exam
Written exam and homework assignments; these will account for 60% and 40% of the final grade, respectively.