Wide scattered spaces and morasses

TL;DR

This paper demonstrates the relative consistency within ZFC that the power set of omega can be arbitrarily large while every sequence of infinite cardinals up to omega_2 can be realized as the cardinal sequence of some locally compact scattered space.

Contribution

It establishes the consistency of large power sets and specific cardinal sequences being realized in locally compact scattered spaces within ZFC.

Findings

2^omega can be arbitrarily large in some models.

Every sequence of infinite cardinals up to omega_2 can be realized.

Consistency results within ZFC for scattered spaces.

Abstract

We show that it is relatively consistent with ZFC that 2^omega is arbitrarily large and every sequence s=(s_i:i<omega_2) of infinite cardinals with s_i<=2^omega is the cardinal sequence of some locally compact scattered space.