# Axiomatic Set Theory (SoSe 2018)

Times and dates for this lecture will be fixed during a preparatory meeting on March 1st at 12:00 in the lecture room of the Kurt Gödel Research Center (Währinger Straße 25, 2nd floor, 02.101, floor plan). If you have any questions or will not be able to participate in the preparatory meeting, please write me an e-mail.

To pass this class you need to take an oral exam. The dates will be fixed by appointment.

Vera Fischer and I will offer a Proseminar (Exercise session) for this lecture. I strongly recommend to also participate in the Proseminar. Times and dates for the Proseminar will also be fixed in the preparatory meeting.

### 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 SoSe 2017) or "Introduction to mathematical logic" (e.g. see WiSe 2017/18) will be helpful.

### References

1. Kunen, K. (2011). Set Theory. College Publications.
2. Jech, T. (2006). Set Theory: The Third Millennium Edition, revised and expanded. Springer Berlin Heidelberg.
3. Halbeisen, L. J. (2011). Combinatorial Set Theory: With a Gentle Introduction to Forcing. Springer London.
4. Schindler, R. (2014). Set Theory: Exploring Independence and Truth. Springer International Publishing.