### Set Theory - M1 - 8EC (2018)

Aim:

To provide the students with a basic knowledge of axiomatic, combinatorial, and descriptive set theory. 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.

Description:

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, a major focus will be descriptive set theory, the study of definable subsets of the real line and their relation to concepts from topology and measure theory (cf. Prerequisites below).

Organization:

The three-hour period will generally be divided into 120 minutes of lectures and a short exercise class.

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, however, assume mathematical maturity, including the naïve use of sets that is very common in mathematics.

Furthermore, in this course, we shall use basic notions and results from Mathematical Logic and Model Theory and we expect students to be familiar with this material.

Students who did not take an introductory course on mathematical logic can find the material in, e.g.,

• Chapters II-VI of Mathematical Logic by Ebbinghaus, Flum, and Thomas,
• Chapter Two of A mathematical introduction to logic by Enderton, Chapter 2 of Introduction to Mathematical Logic by Mendelson, Chapters 3 and 4 of Logic & Structure by van Dalen,
• or in any other introductory textbook on mathematical logic.

The advanced part of the course will focus on descriptive set theory, and for this we expect the students to be familiar with the basics of pointset topology of Euclidean and metric spaces, as well as of the Lebesgue measure of the real line.