## Higher set theory

### Master module V4A3

### Lecturers

- Prof.Dr. Peter Koepke (Lectures)
- Dr. Bernhard Irrgang (Exercises)

### Time and location

Lecture: Monday, 14-16, Wednesday 13-15, Zeichensaal, Wegelerstra&suml;e 10.

Exercises: Monday 10-12, Seminar Room A, Beringstrasse 4, and other times

### Contents

Zermelo-Fraenkel set theory (**ZFC**) axiomatizes the relation "*x* is an element of *y*",
*x ∈ y*. The common mathematical notions (numbers, relations, functions, finite and infinite sequences)
can be formalized by *&isin*-formulas, so that the basic properties of the notion can be proved from the
axioms of **ZFC**. So the Zermelo-Fraenkel set theory provides a foundation for mathematics.
In addition, the system **ZFC** is an axiomatization of the infinite, i.e., of general infinite
numbers and structures. The most important classes of numbers in set theory are the ordinal numbers and the cardinal
numbers. They satisfy remarkable arithmetic and combinatorial laws.

The lecture partially follows the first section of the standard reference *Set Theory -
The Third Millennium Edition* by Thomas Jech, Springer Monographs in Mathematics. We shall cover chapters 1 to 6
and selected material from chapters 7 to 12.