Indescribable cardinals without diamonds
type: Item
instance of: scholarly article
title: Indescribable cardinals without diamonds
zbMATH Open document ID: 0742.03018
DOI: 10.1007/BF01627508
author: Kai Hauser
published in: Archive for Mathematical Logic
publication date: 1992-06-25T00:00:00Z
review text: Recall the combinatorial principle \(\diamondsuit_{\kappa,E}\): There is a sequence \(\langle S_ \alpha: \alpha\in E\rangle\) such that \(\forall\alpha\in E\), \(S_ \alpha\subseteq\alpha\), and \(\forall S\subseteq\kappa\), \(\{\alpha\in E:S_ \alpha=S\cap\alpha\}\) is stationary in \(\kappa\), where \(E\) is a stationary subset of some regular cardinal \(\kappa\). Large cardinals carry \(\diamondsuit\) sequences, for example R. Jensen has shown that \(ZFC\vdash\forall\kappa(\kappa\hbox{ subtle}\to\diamondsuit_{\kappa,Reg\cap\kappa})\) , where \(Reg\) is the class of all regular cardinals. Assuming the consistency of the existence of a \(\Pi^ m_ n\) \((m\geq 1\), \(n\geq 1)\) indescribable cardinal, we construct a model in which \(\diamondsuit_{\kappa,Reg\cap\kappa}\) fails at some indescribable cardinal \(\kappa\). Moreover, the model satisfies the Generalized Continuum Hypothesis. The construction uses iterated forcing and master condition arguments together with an elementary embedding characterization of \(\Pi^ m_ n\) indescribability.
Mathematics Subject Classification ID: 03E55
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E50
zbMATH DE Number: 2023
zbMATH Keywords: combinatorial principle
zbMATH Keywords: indescribable cardinal
zbMATH Keywords: Generalized Continuum Hypothesis
zbMATH Keywords: iterated forcing
zbMATH Keywords: elementary embedding
MaRDI profile type: Publication
cites work: Higher Souslin trees and the generalized continuum hypothesis
cites work: Indescribable cardinals and elementary embeddings
cites work: The indescribability of the order of the indescribable cardinals
full work available at URL: https://doi.org/10.1007/bf01627508
OpenAlex ID: W2020974326
Recommended article: The indescribability of the order of the indescribable cardinals
Recommended article: Indestructibility properties of remarkable cardinals
Recommended article: Cardinality without enumeration
Recommended article: Indescribable cardinals and elementary embeddings
Recommended article: Inaccessible cardinals without the axiom of choice
Recommended article: Successive cardinals with no partial square
Recommended article: Q3581797
Recommended article: Q3773872
Recommended article: Some combinatorial properties of ineffable cardinals
Recommended article: Logic Without Borders
description: scientific article; zbMATH DE number 2023