Stationary cardinals
type: Item
instance of: scholarly article
title: Stationary cardinals
zbMATH Open document ID: 0784.03029
DOI: 10.1007/BF01270466
author: Wenzhi Sun
published in: Archive for Mathematical Logic
publication date: 1994-03-29T00:00:00Z
review text: Let \(\kappa\) be an uncountable cardinal. A subset \(A\subseteq \kappa\) is said to be a 1-club set if \(A\) is stationary and every stationary reflection point belongs to \(A\). It is clear that this definition is a generalization of the definition of closed unbounded sets. It is well known that \(\kappa\) is regular if and only if the set of closed unbounded sets generates a proper \(\kappa\)-complete normal filter. We define that \(\kappa\) is said to be a stationary cardinal if the set of 1-club sets of \(\kappa\) generates a proper \(\kappa\)-complete normal filter, which is called the 1-club filter. It turns out that when \(\kappa\) carries the 1- club filter, it is a moderately large cardinal. It is clear that a stationary cardinal is defined in a similar way that a \(\Pi^ 1_ 1\)- indescribable or a superstationary cardinal is defined. This paper will show that every weakly \(\Pi^ 1_ 1\)-indescribable is a stationary cardinal. Also, every stationary cardinal is a greatly Mahlo cardinal, a reflection cardinal and a cardinal that every stationary set reflects. Moreover, if \(\kappa\) is weakly \(\Pi^ 1_ 1\)-indescribable, then the 1-club filter coincides with the \(\Pi^ 1_ 1\)-indescribable filter. The consistency strength of a stationary cardinal is also observed. We prove it is consistent that there is a stationary cardinal which is not weakly compact. We also prove that the existence of an inaccessible cardinal with stationary set reflection property is independent of that of a stationary cardinal. It is well known that weak ineffability is composed of subtlety and \(\Pi^ 1_ 1\)-indescribability in much the same way that weak compactness is composed of inaccessibility and \(\Pi^ 1_ 1\)- indescribability. Hence subtlety could be regarded as a generalization of inaccessibility. It is well known that if \(\kappa\) is subtle, then \(\lozenge_ \kappa\) is true. And Jensen proved that if \(V=L\), then \(\lozenge_ \kappa\) is true without subtlety [\textit{R. Jens…
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E55
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E45
zbMATH DE Number: 492853
zbMATH Keywords: indescribability
zbMATH Keywords: 1-club set
zbMATH Keywords: stationary reflection point
zbMATH Keywords: closed unbounded sets
zbMATH Keywords: stationary cardinal
zbMATH Keywords: \(\kappa\)-complete normal filter
zbMATH Keywords: 1- club filter
zbMATH Keywords: consistency strength
zbMATH Keywords: inaccessible cardinal
zbMATH Keywords: weak ineffability
zbMATH Keywords: subtlety
reviewed by: Wenchang Sun
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01270466
OpenAlex ID: W4233091920
description: scientific article; zbMATH DE number 492853