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 Goedel's Constructible Universe L and thus will provide a consistency proof for the Axiom of Choice and the Generalized Continuum Hypothesis. The construction of L and the proofs of its properties rely heavily on methods and results from Model Theory, so students should (re)familiarize themselves with these.
Lecturer
KP Hart
- Docent: KP Hart