Set Theory [Fall 2019]
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.
Aim of the course
The aim is to provide the students with a basic knowledge of axiomatic and combinatorial 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.
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 use of permutation models in independence proofs in Set Theory, in particular in proofs of the unprovability of the Axiom of Choice and its consequences.
The three-hour period will generally be divided into 120 minutes of lectures and a short exercise class.
KP Hart and Benedikt Loewe