Creature forcing and large continuum: the joy of halving
type: Item
instance of: scholarly article
title: Creature forcing and large continuum: the joy of halving
zbMATH Open document ID: 1259.03063
author: Jakob Kellner
author: S. Shelah
published in: Archive for Mathematical Logic
publication date: 2012-02-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1003.3425
review text: \textit{M. Goldstern} and the second author [Arch. Math. Logic 32, No. 3, 203--221 (1993; Zbl 0786.03030)] introduced cardinal characteristics of the form \(c^\forall(f,g)\) and \(c^\exists(f,g)\) defined as follows. Let \(f,g\in\omega^{\omega}\), an \(\left( f,g\right) \)-slalom is a sequence \(S=\left\langle S\left( n\right) :n\in\omega\right\rangle \) such that \(S\left( n\right) \subseteq f\left( n\right) \) and \(\left| S\left( n\right) \right| \leq g\left( n\right) \). A family \(\mathcal{S}\) of \(\left( f,g\right) \)-slaloms is a \(\left( \forall,f,g\right) \)-cover, if for all \(r\in\prod_{n\in\omega}f\left( n\right) \) there is \(S\in\mathcal{S}\) such that \(r\left( n\right) \in S\left( n\right) \) for all \(n\in\omega\). Also, a family \(\mathcal{S}\) of \(\left( f,g\right) \)-slaloms is a \(\left( \exists,f,g\right) \)-cover, if for all \(r\in\prod_{n\in\omega}f\left( n\right) \) there is \(S\in\mathcal{S}\) such that \(r\left( n\right) \in S\left( n\right) \) for infinitely many \(n\in\omega\). The minimal sizes of such covers are denoted by \(c_{f,g}^{\forall}\) and \(c_{f,g}^{\exists}\), respectively. In the paper under review the authors continue their work initiated in [J. Symb. Log. 74, No. 1, 73--104 (2009; Zbl 1183.03035)]. Here they introduce a new mixed-limit creature forcing construction to show that it is consistent that \(c^\forall(f,g)=c^\exists(f,g)=\kappa_\epsilon\) for continuum many pairwise different cardinals \(\kappa_\epsilon\) and suitable pairs \((f_\epsilon,g_\epsilon)\).
Mathematics Subject Classification ID: 03E17
Mathematics Subject Classification ID: 03E35
zbMATH DE Number: 6005017
zbMATH Keywords: slaloms
zbMATH Keywords: creature forcing
zbMATH Keywords: cardinal characteristics
reviewed by: Fernando Hernández-Hernández
MaRDI profile type: Publication
OpenAlex ID: W2102578166
arXiv ID: 1003.3425
cites work: Q4848568
cites work: Many simple cardinal invariants
cites work: Even more simple cardinal invariants
cites work: Decisive creatures and large continuum
cites work: Norms on possibilities. I. Forcing with trees and creatures
DOI: 10.1007/S00153-011-0253-8
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description: scientific article