Annotation
We will focus on a particular topic which is on the frontier of mathematics and set theory and will map its development. In 2022/23, we will focus on LARGE CARDINALS. These are uncountable regular cardinals which feature some extra combinatorial properties such as the existence of a non-trivial kappa-complete ultrafilter (measurable cardinal), or the fact that every kappa-tree has a cofinal branch (with inaccessibility this characterizes weak compactness).
This will be an introductory survey requiring just basic knowledge of set theory. We discuss the development of these notions, connected as they often are to infinitary logic and model theory. We will introduce some basic definitions: inaccessible, Mahlo, weakly compact, Ramsey and measurable cardinals, and show some implications between them. We will also discuss the role of large cardinals as a scale to measure the consistency strength of ordinary mathematical statements, which at the first glance have nothing to do with set theory and large cardinals (such as the existence of sigma-complete measures extending the Lebesgue measure).
We will also mention philosophical aspects related to large cardinals: it was Goedel who first suggested that large cardinal axioms are natural candidates for extending the standard set theory ZFC in order to at least partially overcome the phenomenon of incompleteness. By extending ZFC by large cardinal axioms we can get a natural theory which is strictly stronger than ZFC, and can thus prove more interesting mathematical statements.