On almost precipitous ideals
type: Item
instance of: scholarly article
title: On almost precipitous ideals
zbMATH Open document ID: 1203.03063
author: Asaf Ferber
author: Moti Gitik
published in: Archive for Mathematical Logic
publication date: 2010-04-15T00:00:00Z
review text: A regular uncountable cardinal \(\kappa\) is \textit{almost precipitous} if for each ordinal \(\tau < {(2^\kappa)}^+\), there is a \(\kappa\)-complete ideal \(I\) over \(\kappa\) with the property that every generic ultrapower of \(I\) is well-founded up to the image of \(\tau\). The paper presents a wealth of results concerning almost precipitousness and related notions (semi-precipitousness, precipitousness and pseudo-precipitousness). Here are three of these results: {\parindent=7mm \begin{itemize}\item[(A)] ``There exists an almost precipitous cardinal'' and ``There exists an almost precipitous cardinal with normal ideals witnessing its almost precipitousness'' are equiconsistent; \item[(B)] Assume \(0^\#\), and let \(\eta\) be a regular cardinal in \(L\). Then there is a generic extension \(L[G]\) of \(L\) such that (a) \(L[G]\) and \(L\) have the same cardinals less than or equal to \(\eta\), and (b) in \(L[G]\), \(\eta^+\) is almost precipitous as witnessed by normal filters; \item[(C)] If there are class many Ramsey cardinals, then any uncountable cardinal is almost precipitous as witnessed by normal filters. \end{itemize}}
reviewed by: Pierre Matet
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E35
zbMATH DE Number: 5693389
zbMATH Keywords: generic ultrapowers
zbMATH Keywords: well foundedness
zbMATH Keywords: almost precipitous ideals
MaRDI profile type: Publication
full work available at URL: s00153-009-0173-z
OpenAlex ID: W2079317904
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DOI: 10.1007/S00153-009-0173-Z
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description: scientific article; zbMATH DE number 5693389