Axiomatic Set Theory (SoSe 2018)


The lecture takes place on Thursdays from 11:30am to 1:45pm in the lecture room of the Kurt Gödel Research Center (Währinger Straße 25, 2nd floor, 02.101, floor plan).

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

To pass this class you need to take an oral exam. The duration of each exam will be around 30min and the dates for the exams in this semester are Additional dates for exams will be offered for the next semester, probably during the first two weeks of October, beginning of December, and at the end of January. To sign up for an exam you have to write me an e-mail with your Matrikelnummer and preferred date. The time slots for the exams will be assigned on a first come first serve basis.


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.


Proseminar

Vera Fischer and I will offer a Proseminar (Exercise session) for this lecture. I strongly recommend to also participate in the Proseminar. The Proseminar takes place on Fridays from 1pm to 2:30pm in the lecture room of the Kurt Gödel Research Center (Währinger Straße 25, 2nd floor, 02.101, floor plan). To be graded for the Proseminar you have to regularly attend and present solutions on the board at least two times during the semester. To practice your mathematical writing skills, we in addition expect you to occasionally submit written solutions for exercises.

Exercises

Problem set 1: 09.03.2018
Problem set 2: 17.03.2018
Problem set 3: 23.03.2018
Problem set 4: 14.04.2018
Problem set 5: 20.04.2018
Problem set 6: 27.04.2018
Problem set 7: 04.05.2018
Problem set 8: 11.05.2018
Problem set 9: 18.05.2018

The exercises can also be found here and on moodle.


References

  1. Kunen, K. (2011). Set Theory. College Publications.
  2. Schindler, R. (2014). Set Theory: Exploring Independence and Truth. Springer International Publishing.
  3. 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.
  1. Halbeisen, L. J. (2011). Combinatorial Set Theory: With a Gentle Introduction to Forcing. Springer London.