Algebra Seminar talk

2018-04-27 (10:15)
Johannes Schürz
Reell messbare Kardinalzahlen

Vitali's famous counterexample shows that there cannot exist a total, translation invariant measure on $[0,1]$. Naturally, the question arises what happens if one drops the requirement of translation invariance, i.e. does there exist a total measure $\mu$ extending the Lebesgue measure?

Surprisingly, this naive question gives rise to a branch of set theory, the so called ‘Large Cardinals’. These are cardinals whose existence would imply the consistency of ZFC. In my talk I want to define the required notions and eventually prove the following result by Ulam: “If such a measure exists, then $2^{\aleph_0} \ge$ the least weakly inaccessible cardinal”. I also want to mention some interesting results by Solovay.