Historic forcing for Depth
type: Item
instance of: scholarly article
title: Historic forcing for Depth
zbMATH Open document ID: 0982.03029
author: Andrzej Rosłanowski
author: S. Shelah
published in: Colloquium Mathematicum
publication date: 2001-06-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0006219
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 06E05
zbMATH DE Number: 1605332
zbMATH Keywords: cardinal invariants
zbMATH Keywords: Boolean algebras
zbMATH Keywords: depth
zbMATH Keywords: historic forcing
MaRDI profile type: Publication
OpenAlex ID: W2964209286
arXiv ID: math/0006219
DOI: 10.4064/CM89-1-7
review text: The depth of a Boolean algebra \({\mathfrak B}\), \(\text{Depth}({\mathfrak B}\)), is the supremum of all cardinalities of well-ordered subsets \({\mathfrak B}\). NEWLINENEWLINENEWLINEIn this paper the authors show consistently that for regular cardinals \(\theta, \lambda\) with \(\theta <\lambda\) there exists a Boolean algebra \({\mathfrak B}\) with \(|{\mathfrak B}|=\lambda^+\) such that for every subalgebra \({\mathfrak B}'\subset {\mathfrak B}\) with \(|{\mathfrak B}'|= \lambda^+\), we have \(\text{Depth} ({\mathfrak B}') =\theta\). This partially answers a question of Monk. NEWLINENEWLINENEWLINEThe result is obtained by means of a special forcing, the so-called historic forcing, which has been developed by Shelah some years ago and has been used by him already to obtain consistency results in partition calculus.
reviewed by: Martin Weese
description: scientific article; zbMATH DE number 1605332