Hausdorff gaps and towers in P(\omega)/Fin

TL;DR

This paper investigates uncountable chains called towers in the power set of natural numbers modulo finite sets, exploring their properties, existence in different models, and their relation to gaps in set theory.

Contribution

It introduces and analyzes Hausdorff and Suslin towers, providing examples of indestructible gaps and suggesting directions for a structural theory of towers.

Findings

Existence of Hausdorff and Suslin towers in various models

Construction of indestructible gaps not equivalent to Hausdorff gaps

Potential development of a structure theory for towers

Abstract

We define and study two classes of uncountable $\subseteq^*$-chains: Hausdorff towers and Suslin towers. We discuss their existence in various models of set theory. Then, some of the results and methods are used to provide examples of indestructible gaps not equivalent to a Hausdorff gap. Also, we indicate possible ways of developing a structure theory for towers.