Set Theory Group

Abstract: Ever since Cohen invented forcing and showed the independence of the continuum hypothesis (CH), the powerset function of infinite cardinals has been a central theme in modern set theory. In this talk we will focus of the behavior of the powerset function of singular cardinals. A cardinal $\kappa$ is singular if it can be written as the union of less than $\kappa$ many sets, each of size less than $\kappa$. The singular cardinal hypothesis (SCH) is an analogue of CH for singular cardinals. While it is possible to violate it by forcing, such constructions are generally difficult to obtain and can be quite intricate. We will present some recent results on the relationship between violating SCH and the combinatorial properties of the singular cardinal.