Cofinal types on ${\omega }_2$

Abstract

In this paper we start the analysis of the class ${\mathcal{D}}_{{\aleph }_2}$, the class of cofinal types of directed sets of cofinality at most ${\aleph }_2$. We compare elements of ${\mathcal{D}}_{{\aleph }_2}$ using the notion of Tukey reducibility. We isolate some simple cofinal types in ${\mathcal{D}}_{{\aleph }_2}$, and then proceed to show which of these types have an immediate successor in the Tukey ordering of ${\mathcal{D}}_{{\aleph }_2}$.