## Models of Set Theory I. (V4A8)

### Lecturers

- PD Dr. Philipp Lücke

### Time and place

Lecture: Monday 14:15-16:00 und Wednesday 12:15-14:00, Seminarraum 1.008, Endenicher Allee 60. Start: 01. April.Problem classes: Wednesday 10:15-12:00, Seminarraum N0.008, Endenicher Allee 60. Start: 10. April.

### Content

Cantor's*Continuum Hypothesis*(CH) is the statement that every uncountable set of real numbers has the same cardinality as the set of all real numbers. The validity of CH is one of the most central problems in set theory and it was considered so important by David Hilbert that he placed it first on his famous list of open problems to be faced by the 20

^{th}century. In this lecture course, we will present groundbreaking results of Kurt GĂ¶del and Paul Cohen that can be used to show that CH is independent of the standard axiomatization of mathematics provided by the

*Zermelo-Fraenkel axioms of Set Theory*together with the

*Axiom of Choice*(ZFC).

The lecture course will cover the following topics:

- Transitive models of ZFC
- The inner model HOD
- The constructible universe
- Forcing
- The independence of the Continuum Hypothesis
- The independence of the Axiom of Choice

### Problems

- Series 1 (due Monday, April 08).
- Series 2 (due Monday, April 15).
- Series 3 (due Tuesday, April 23).
- Series 4 (due Monday, April 29).
- Series 5 (due Monday, May 06).
- Series 6 (due Monday, May 13).
- Series 7 (due Monday, May 20).
- Series 8 (due Monday, May 27).
- Series 9 (due Monday, June 03. Problem 36 corrected).
- Series 10 (due Monday, June 17).
- Series 11 (due Monday, June 24).
- Series 12 (due Monday, July 01).

### Oral Exams

**1. Date**: 15.-17. July 2019.**2. Date**: 23.-27. September 2019.

### Literature

- Kenneth Kunen:
**Set theory - An introduction to independence proofs**. Studies in Logic and the Foundations of Mathematics, 102. North-Holland Publishing Co., Amsterdam-New York, 1980. - Ralf Schindler:
**Set theory - Exploring independence and truth**. Universitext. Springer, Cham, 2014. - Thomas Jech:
**Set theory - The third millennium edition, revised and expanded**. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003.

Last changed: 18.01.2019