Axiomatic Set Theory

(SoSe 2020)

(SoSe 2020)

The lecture takes place on Thursdays from 12:00pm to 3:15pm (with a 15min break approx. from 13:30pm to 13:45pm) in the lecture room of the Kurt Gödel Research Center (Augasse 2-6, 5th floor, green area D, seminar room 5.48). Lectures start on March 5th. Please check u:find to see in which weeks there are no lectures.

Due to the COVID-19 related closure of the University of Vienna this class takes place virtually on moodle. Please, check out the moodle page for further information.

Details about the contents of each lecture can be found here.

To pass this class you need to take an oral exam. The exams will take place virtually, you can find more information on the moodle page of this class.

### Contents

This lecture will be an introduction to set theory, in particular to independence proofs. The goal is to establish the independence of the continuum hypothesis. We will start from the ZFC axioms and introduce ordinals and cardinals. Then we will define Gödel's constructible universe $L$ and show that it is a model of ZFC and GCH, the generalized continuum hypothesis. Furthermore, we will introduce measurable cardinals and show that they cannot exist in $L$. If time allows, we will discuss variants $L[U]$ of $L$ which allow the existence of a measurable cardinal. Finally, we will introduce Cohen's forcing technique and show that there is a model of ZFC in which the continuum hypothesis does not hold.Some familiarity with the contents of the lecture "Grundzüge der mathematischen Logik" (e.g. see WiSe 2019/20) or "Introduction to mathematical logic" (e.g. see SoSe 2019) will be helpful.

### Proseminar

Jonathan Schilhan will offer a Proseminar (Exercise session) for this lecture. I strongly recommend to also participate in the Proseminar.### References

- Kunen, K. (2011).
*Set Theory*. College Publications. - Schindler, R. (2014).
*Set Theory: Exploring Independence and Truth*. Springer International Publishing. - Jech, T. (2006).
*Set Theory: The Third Millennium Edition, revised and expanded*. Springer Berlin Heidelberg.

For further studies I also recommend the following book, but most material in this book will be beyond what we will be able to cover this semester.

- Halbeisen, L. J. (2011).
*Combinatorial Set Theory: With a Gentle Introduction to Forcing*. Springer London.