Projects in Previous Years

2nd Semester 2015/16: Measurable Cardinals

Lorenzo Galeotti (ILLC) and Hugo Nobrega (ILLC)
If you are interested in this project, please contact the instructors by email.
Large cardinals are certain types of objects whose existence cannot be proved using the standard axioms of set theory (ZFC), and are in fact very useful for measuring how much, beyond ZFC, one needs to assume in order to be able to prove certain desired results. Measurable cardinals, in particular, have been described as "the most prominent of all large cardinal hypotheses" [1, p. XVI].

In this project we will study these cardinals, starting from the basics of measure theory and filters and going all the way to the proof that there are no measurable cardinals in Goedel's constructible universe L. We will also see different characterizations of these objects coming from different areas of mathematics and logic.

Lectures will happen twice a week in the period of 30 May -- 24 June 2016.
A basic set theory course (up to cardinal arithmetic and the definition of the constructible universe), a basic model theory course (up to the ultrapower construction and Los's theorem), and general mathematical maturity.
Homework and an oral exam.
We will mainly use the following standard reference text.

[1] Akihiro Kanamori. The Higher Infinite (2nd ed). Springer, 2003.