### Set Theory - M1 - 8EC

Prerequisites
Prerequisites. The course is a combination of an introductory and an advanced
course in set theory. As a consequence, no prior knowledge of axiomatic set
theory is assumed. We shall assume familiarity with the naïve use of sets
that is common in mathematics.

We shall use basic notions and results from mathematical logic and expect
students to be familiar with this material. Knowledge of basic model theory,
i.e., compactness arguments, Löwenheim-Skolem theorems, etc.
(as usually taught in a course on mathematical logic) will be indispensible.
Chapter 2 of Introduction to Mathematical Logic by E. Mendelson covers the
material that we will use freely.

This is a mathematics course at the Master's level and is primarily aimed at
students with an undergraduate degree in mathematics. We expect the
mathematical maturity that comes with such a degree, but hardly any specific
knowledge about mathematics (other than mathematical logic as mentioned in
the last paragraph). Thus, in theory, the course is accessible to students
from a non-mathematical background, as long as they have the required
mathematical maturity. Nevertheless, most of the illustrative examples will
come from mathematics, so the latter students should be willing to read up
on the areas where the examples come from.

Aim of the course
The aim is to provide the students with a basic knowledge of axiomatic and

combinatorial set theory, model constructions in set theory, and to prepare the
students for research in set theory and for using set theory as a tool in
mathematical areas such as general topology, algebra and functional analysis.

The course will start with a brief introduction to axiomatic set theory,
the model theory of set theory (including simple independence results),
and the basic theory of ordinals and cardinals. The second part of
the course will be devoted to more advanced topics in set theory.

This year, the focus of the advanced topics will be the Shelah's PCF Theory,
in particular the by now well-known inequality
$2^{\aleph_\omega}<\aleph_{\omega_4} in case \aleph_\omega$ is a strong limit.

Lecturer

KP Hart