Cofinal types on $\omega_2$

TL;DR

This paper investigates the structure of cofinal types of directed sets with cofinality up to , using Tukey reducibility to classify and analyze their ordering and immediate successors.

Contribution

It introduces the class  of cofinal types, identifies simple types within it, and examines the Tukey ordering and immediate successors among these types.

Findings

Identification of simple cofinal types in 

Characterization of immediate successors in the Tukey order

Comparison framework for cofinal types using Tukey reducibility

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 $\mathcal D_{\aleph_2}$.