Ineffability of \(\mathcal{P}_{\kappa} \lambda\) for \(\lambda\) with small cofinality
type: Item
instance of: scholarly article
title: Ineffability of \(\mathcal{P}_{\kappa} \lambda\) for \(\lambda\) with small cofinality
zbMATH Open document ID: 1154.03034
author: Toshimichi Usuba
published in: Journal of the Mathematical Society of Japan
publication date: 2008-09-18T00:00:00Z
review text: This paper is about ineffability, the Shelah property, and indescribability of \(\mathcal{P}_\kappa\lambda\). The author proves many interesting results about them. Let \(X\) be a subset of \(\mathcal{P}_\kappa\lambda\). \(X\) is said to be ineffable (respectively almost ineffable) if for every \(\langle a_x : x\in X\rangle\) with \(a_x\subseteq x\), there exists an \(A\subseteq\lambda\) such that \(\{x\in X : A\cap x=a_x\}\) is stationary (respectively unbounded). \(X\) has the Shelah property if for every \(\langle f_x : x\in X\rangle\) with \(f_x:x\rightarrow x\), there exists an \(f:\lambda\rightarrow\lambda\) such that for every \(y\in\mathcal{P}_\kappa\lambda\), \(\{x\in X : f\restriction y=f_x\restriction y\}\) is unbounded. Define \(\text{NIn}_{\kappa \lambda}=\{X\subseteq\mathcal{P}_\kappa\lambda : X\text{ is not ineffable}\}\), \(\text{NAIn}_{\kappa \lambda}=\{X\subseteq\mathcal{P}_\kappa\lambda : X\text{ is not almost ineffable}\}\), and \(\text{NSh}_{\kappa \lambda}=\{X\subseteq\mathcal{P}_\kappa\lambda : X\text{ does not have the Shelah property}\}\). All of them form ideals over \(\mathcal{P}_\kappa\lambda\). \textit{D. M. Carr} [Z. Math. Logik Grundlagen Math. 31, 393--401 (1985; Zbl 0559.03033)] proved \(\text{NSh}_{\kappa \lambda}\subseteq\text{NIn}_{\kappa \lambda} \subseteq\text{NAIn}_{\kappa \lambda}\). Here are some of the results the author proves: (1) these three ideals are always strongly normal (\textit{D. M. Carr} [loc. cit.] proved this assuming \(\text{cf}(\lambda)\geq\kappa\), but it was open without the assumption); (2) if \(\mathcal{P}_\kappa\lambda\) has the Shelah property and \(\text{cf}(\lambda)<\kappa\), then \(\lambda^{{<}\kappa}=\lambda^+\); (3) if \(\lambda\) is strong limit with \(\text{cf}(\lambda)<\kappa\), then \(\text{NSh}_{\kappa \lambda}=\text{NIn}_{\kappa \lambda} =\text{NAIn}_{\kappa \lambda}\); (4) if \(\text{cf}(\lambda)\geq\kappa\), then, except the trivial case, \(\text{NSh}_{\kappa \lambda}\subsetneq\text{NIn}_{\k…
reviewed by: Tetsuya Ishiu
Mathematics Subject Classification ID: 03E55
Mathematics Subject Classification ID: 03E05
zbMATH DE Number: 5345244
zbMATH Keywords: \(\mathcal{P}_\kappa \lambda\)
zbMATH Keywords: ineffable
zbMATH Keywords: almost ineffable
zbMATH Keywords: Shelah property
zbMATH Keywords: indescribable
zbMATH Keywords: ideals
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/06030935
OpenAlex ID: W2594332275
cites work: Saturation of fundamental ideals on \({\mathcal P}_ \kappa \lambda\)
cites work: Combinatorial characterization of \(\Pi_1^1\)-indescribability in \(P_\kappa\lambda\)
cites work: Notes on subtlety and ineffability in \(P_\kappa\lambda\)
cites work: Q4071748
cites work: Shelah's pcf theory and its applications
cites work: The Minimal Normal Filter on P κ λ
cites work: Pxδ‐Generalizations of Weak Compactness
cites work: The structure of ineffability properties of \(P_{\kappa}\lambda\)
cites work: On the existence of strongly normal ideals over \(P_{\kappa}\lambda\)
cites work: Q5557903
cites work: The entire NS ideal on can be precipitous
cites work: Some combinatorial problems concerning uncountable cardinals
cites work: On ideals and stationary reflection
cites work: Some partition relations for ideals on \(P_{\kappa}\lambda\)
cites work: Q4312353
cites work: Q4315890
cites work: Q4078034
cites work: Rowbottom-Type Properties and a Cardinal Arithmetic
DOI: 10.2969/JMSJ/06030935
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description: scientific article; zbMATH DE number 5345244