Some results on the nonstationary ideal
type: Item
instance of: scholarly article
title: Some results on the nonstationary ideal
zbMATH Open document ID: 0849.03042
DOI: 10.1007/BF02762071
published in: Israel Journal of Mathematics
publication date: 1996-11-04T00:00:00Z
review text: The author investigates further the consistency strengths of hypotheses on \(\text{NS}_\kappa\) (the nonstationary ideal over cardinal \(\kappa\)) and \(\text{NS}^\lambda_\kappa\) (the nonstationary ideal over \(\kappa\) restricted to sets of ordinals of cofinality \(\lambda\)); the three properties studied are precipitousness, presaturatedness and saturatedness. Among the many interesting results in the paper are the following: (1) The exact strength of ``\(\text{NS}^\lambda_{\mu^+}\) is precipitous for a regular \(\mu > \max (\lambda, \aleph_1)\)'' is an \((\omega,\mu)\)-repeat point; (2) The exact strength of ``\(\text{NS}_\kappa\) is presaturated over inaccessible \(\kappa\)'' is an up-repeat point; (3) ``\(\text{NS}_\kappa\) is saturated over inaccessible \(\kappa\)'' implies an inner model with \(\exists \alpha (o(\alpha) = \alpha^{++})\). The notion of a repeat point was originally introduced by \textit{L. B. Radin} [Ann. Math. Logic 22, 243-261 (1982; Zbl 0502.03028)] in his paper introducing what has since become known as Radin forcing. Related notions are defined in the paper under review. For example, with \(\vec {\mathcal F}\) a coherent sequence of ultrafilters, an ordinal \(\alpha\) less than the length of \(\vec {\mathcal F}\) is defined to be an up-repeat point if for each set \(A\) in \(\vec {\mathcal F} (\kappa,\alpha)\) there is \(\beta > \alpha\) such that \(A \in \vec{\mathcal F} (\kappa,\beta)\).
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E55
zbMATH DE Number: 836156
zbMATH Keywords: precipitous ideal
zbMATH Keywords: presaturated ideal
zbMATH Keywords: saturated ideal
zbMATH Keywords: consistency strength
zbMATH Keywords: nonstationary ideal
zbMATH Keywords: inner model
zbMATH Keywords: repeat point
zbMATH Keywords: forcing
zbMATH Keywords: up-repeat point
author: Moti Gitik
reviewed by: Neil H. Williams
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 836156