MATH 7820: Seminar in Logic: Combinatorics of singular cardinals (Spring 2010)

Instructor: Justin Moore

The logic seminar has two components — a long-running topic and several shorter groups of lectures on topics related to the interests of students, professors, and visitors. Those enrolled in the seminar will be expected to present lectures on papers related to the topic once the basics are developed.

The focus of the topic portion of the seminar will be to expore singular cardinals and their combinatorial properties. A cardinal $\kappa$ is singular if $\kappa = \sup_{i \in I} \kappa_i$ where both I and each $\kappa_i$ have cardinality less than $\kappa$. $\aleph_\omega = \sup_n \aleph_n$ is the first singular cardinal and the one which is most studied. While cardinal exponentiation of regular (i.e. non-singular) cardinals has been understood since the beginnings of forcing, the behavior of cardinal exponentiation at singular cardinals is still an area of active research in set theory. Shelah proved that there are highly non trivial limitations on cardinal exponentiation of singular cardinals, culminating in his celebrated result that if $2^{\aleph_n} < \alpha_\omega$ for all $n < \omega$, then $2^{\aleph_\omega} < \aleph_{\omega_4}$. In proving this, he developed the study of "possible cofinalities" of ultrapowers. The topic this semester will be to develop Shelah's pcf theory, and the theory of singular (and regular) cardinals in general, from the ground up. Basic familiarity with cardinals is a sufficient prerequisite.