1st Semester 2019/20: Selected Topics in Set Theory
This project is in seminar form, with students giving presentations on advanced topics in set theory. This time, the topic will be "Gödel's Constructible Universe".
In 1938 Gödel constructed a model of set theory, known as the Constructible Universe L, in which the Axiom of Choice (AC) and the Generalized Continuum Hypothesis (GCH) are satisfied, proving that neither AC nor GCH could be refuted on the basis of ZF. The model L is generally seen as a "minimal model of set theory", and has since been shown to satisfy many other interesting properties.
In this project, we cover the basic theory of Gödel's Constructible Universe L and related topics. In particular:
- Models of set theory and absoluteness
- Reflection Theorems
- Basic properties of the constructible universe L
- AC in L
- GCH in L
- Depending on the number of participants, we may cover additional topics about L, such as: diamond and combinatorial principles, Suslin trees, definable well-order of the reals, regularity properties for projective sets.
At the beginning of January we will decide the precise topics to cover and assign a section to the students. The students will study it independently in the course of several weeks (with personal guidance if needed), and give a 2h presentation at the end of January. To conclude, each student will also hand in a short written summary of the material they presented.
The MasterMath "Set Theory" course or equivalent.
Assessment is based on the presentations and the written reports, on a "Pass"/"Fail" basis.
Thomas Jech, "Set Theory" (third Millennium Edition), Springer-Verlag 2003
Kenneth Kunen, "Set Theory" (2013 Edition)