A consistency theorem for cardinal sequences of length $< \omega_3$

TL;DR

This paper proves the consistency of certain cardinal sequences for scattered Boolean spaces of length less than , extending the understanding of their possible structures within set theory.

Contribution

It establishes a new consistency result for cardinal sequences of scattered Boolean spaces with sequences of length less than , under specific conditions.

Findings

Shows the consistency of specified cardinal sequences for scattered Boolean spaces

Extends known results to sequences of length less than 

Provides a framework for constructing such spaces under set-theoretic assumptions

Abstract

We prove that if $\lambda$ is a fixed uncountable cardinal and $f = \langle \ka_{\al} : \al < \delta \rangle$ is a sequence of infinite cardinals where $\delta < \omega_3$ and $\ka_{\al}\in \{\om,\lambda\}$ for each $\al < \delta$ in such a way that $f^{-1}\{\om\}$ is $\om_2$-closed in $\delta$, then it is consistent that there is a scattered Boolean space whose cardinal sequence is $f$.