Cardinal invariants of the continuum and combinatorics on uncountable cardinals
type: Item
instance of: scholarly article
title: Cardinal invariants of the continuum and combinatorics on uncountable cardinals
zbMATH Open document ID: 1112.03046
author: Jörg Brendle
published in: Annals of Pure and Applied Logic
publication date: 2007-02-02T00:00:00Z
review text: The author explores the connection between combinatorial principles on the reals, like stick and club, on the one hand, and the combinatorics of sets of reals on the other hand. He starts by showing the following fact: Assume that the null ideal \(\mathcal N\) is \(< \mathfrak{c}\)-additive. Then Martin's axiom holds for any \(\mathbb{C}_{\lambda}\), where \(\mathbb{C}\) denotes the corresponding Cohen algebra. Next, he deals with Gross spaces. It is an old result of Baur and Gross, that Gross spaces exist under CH. On the other hand it was shown by Shelah and Spinas that it is consistent that \(\mathfrak{c} = \aleph_2\) and there are no Gross spaces over any finite field, and it was shown by Shelah that it is consistent that \(\mathfrak{c} = \aleph_3\) and there are no Gross spaces over any at most countable field. Here, the author shows that under \(\mathfrak{c} = \aleph_2\) there is a Gross space over every countable field. It is shown that it is consistent that \(\mathfrak{c} = \aleph_2\), \(\clubsuit\) holds and the union of less than \(\mathfrak{c}\) many null sets does not cover the real line. There are results concerning Suslin trees and mad families. So it is shown: Assume that \(\clubsuit\) holds and the meagre ideal \(\mathcal M\) has a base of size \(\aleph_1\). Then there is a Suslin tree. Assume \(\clubsuit\) and there is a dominating family of size \(\aleph_1\). Then there is a mad family of size \(\aleph_1\). In two appendices the author solves problems of Fuchino, Shelah and Soukup, and of Kraszewski, respectively. So he shows that it is consistent that {cov}(\(\mathcal M\))\(=\aleph_2\) and \(\clubsuit_S\) holds for all stationary \(S \subseteq \omega_1\).
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E17
Mathematics Subject Classification ID: 03E50
zbMATH DE Number: 5121368
zbMATH Keywords: cardinal invariants of the continuum
zbMATH Keywords: Martin's axiom
zbMATH Keywords: meagre ideal
zbMATH Keywords: null ideal
zbMATH Keywords: Cohen forcing
zbMATH Keywords: random forcing
zbMATH Keywords: Gross space
zbMATH Keywords: reaping number
zbMATH Keywords: stick principle
zbMATH Keywords: club principle
zbMATH Keywords: Suslin tree
zbMATH Keywords: dominating family
zbMATH Keywords: mad family
reviewed by: Martin Weese
MaRDI profile type: Publication
full work available at URL: j.apal.2006.05.003
OpenAlex ID: W2025373564
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Cardinal invariants of the continuum and combinatorics on uncountable cardinals (DOI)
DOI: 10.1016/J.APAL.2006.05.003