# A consistency result on long cardinal sequences

## TL;DR

This paper proves a consistency result showing that for any regular cardinal and certain ordinals, it is possible to have a large continuum and construct locally compact scattered spaces with prescribed cardinal sequences.

## Contribution

It establishes a new consistency result linking large continuum hypotheses with the existence of specific locally compact scattered spaces.

## Key findings

- Consistency of large continuum with prescribed cardinal sequences
- Construction of locally compact scattered spaces with specific properties
- Extension of previous results on cardinal sequences and topology

## Abstract

For any regular cardinal $\kappa$ and ordinal $\eta<\kappa^{++}$ it is consistent that $2^{\kappa}$ is as large as you wish, and every function $f:\eta \to [\kappa,2^{\kappa}]\cap Card$ with $f(\alpha)=\kappa$ for $cf(\alpha)<\kappa$ is the cardinal sequence of some locally compact scattered space.