Monotone hulls for \(\mathcal N \cap\mathcal M\)
type: Item
instance of: scholarly article
title: Monotone hulls for \(\mathcal N \cap\mathcal M\)
zbMATH Open document ID: 1340.03016
author: Andrzej Rosłanowski
author: S. Shelah
published in: Periodica Mathematica Hungarica
publication date: 2015-01-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/1007.5368
review text: Using the method of decisive creatures (see [\textit{J. Kellner} and \textit{S. Shelah}, J. Symb. Log. 74, No. 1, 73--104 (2009; Zbl 1183.03035)]) we show the consistency of ``there is no increasing \(\omega_2\)-chain of Borel sets and non\((\mathcal{N}) =\) non\((\mathcal{M}) =\) non\((\mathcal{N} \cap \mathcal{M}) = \omega_2 = 2^{\omega}\)''. Hence, consistently, there are no monotone Borel hulls for the ideal \(\mathcal{M} \cap\mathcal{N}\). This answers \textit{M. Balcerzak} and \textit{T. Filipczak} [Math. Log. Q. 57, No. 2, 186--193 (2011; Zbl 1235.03073)]. Next we use finite support iteration of ccc forcing notions to show that there may be monotone Borel hulls for the ideals \(\mathcal{M}, \mathcal{N}\) even if they are not generated by towers.
reviewed by: Martin W. Bunder
Mathematics Subject Classification ID: 03E17
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E15
zbMATH DE Number: 6384178
zbMATH Keywords: \(\omega_2\)-chain
zbMATH Keywords: Borel sets
zbMATH Keywords: decisive creatures
MaRDI profile type: Publication
OpenAlex ID: W1982933075
arXiv ID: 1007.5368
cites work: On monotone hull operations
cites work: Q4844091
cites work: Hechler's theorem for the meager ideal
cites work: Coloring ordinals by reals
cites work: Hechler's theorem for the null ideal
cites work: Transfinite sequences of continuous and Baire class 1 functions
cites work: Can we assign the Borel hulls in a monotone way?
cites work: Decisive creatures and large continuum
cites work: Changing cardinal characteristics without changing \(\omega\)-sequences or cofinalities
cites work: Norms on possibilities. I. Forcing with trees and creatures
cites work: Around cofin
cites work: Q4523075
cites work: Was Sierpiński right? IV
cites work: The null ideal restricted to some non-null set may be \aleph1-saturated
cites work: The cofinality spectrum of the infinite symmetric group
DOI: 10.1007/S10998-014-0042-3
description: scientific article