The least strongly compact can be the least strong and indestructible
type: Item
instance of: scholarly article
title: The least strongly compact can be the least strong and indestructible
zbMATH Open document ID: 1115.03073
author: Arthur W. Apter
published in: Annals of Pure and Applied Logic
publication date: 2007-02-02T00:00:00Z
review text: The following three theorems are proved. (A) Suppose \(V\vDash \text{ZFC}+ \text{GCH}+\kappa\) is supercompact. Then, in some forcing extension, (a) \(\kappa\) is both the least strongly compact and least strong cardinal, and (b) each of \(\kappa\)'s strong compactness and strongness is indestructible under \(<\!\kappa\)-directed closed partial orderings which are also \(\kappa\)-strategically closed. (B) Suppose \(V\vDash \text{ZFC}+ \text{GCH}+\kappa_1\) is supercompact \(+\,\kappa_2 < \lambda\) and \(\kappa_2\) and \(\lambda\) are the smallest cardinals above \(\kappa_1\) such that \(\kappa_2\) is \(\lambda\)-supercompact and \(\lambda\) is inaccessible. There is then a partial ordering \(\mathbb{P}\in V\) such that, in \((V_\lambda)^{V^\mathbb{P}},\) (a) \(\kappa_1\) and \(\kappa_2\) are both the first two strongly compact and first two strong cardinals, (b) each of \(\kappa_1\)'s strong compactness and strongness is indestructible under \(<\!\kappa_1\)-directed closed partial orderings which are also \(\kappa_1\)-strategically closed, and (c) \(\kappa_2\)'s strongness, but not its strong compactness, is indestructible under \(\kappa_2\)-strategically closed forcing which is also \(<\!\kappa_1\)-directed closed. (C) Suppose \(V\vDash\text{ZFC}+ \text{GCH}+\kappa\) is supercompact with no Mahlo cardinal above it. Then, in some forcing extension, (a) \(\kappa\) is both the least strongly compact and least strong cardinal, (b) there is no Mahlo cardinal above \(\kappa\), (c) \(\kappa\)'s strong compactness is indestructible under \(<\!\kappa\)-directed closed forcing, and (d) \(\kappa\)'s strongness is indestructible under \(\kappa^+\)-weakly closed forcing notions satisfying the Prikry condition.
reviewed by: Pierre Matet
Mathematics Subject Classification ID: 03E55
Mathematics Subject Classification ID: 03E35
zbMATH DE Number: 5121367
zbMATH Keywords: strongly compact cardinal
zbMATH Keywords: strong cardinal
zbMATH Keywords: indestructibility
zbMATH Keywords: Prikry forcing
zbMATH Keywords: Radin forcing
MaRDI profile type: Publication
full work available at URL: j.apal.2006.05.002
OpenAlex ID: W2033514078
cites work: Some remarks on indestructibility and Hamkins' lottery preparation
cites work: Strong compactness, measurability, and the class of supercompact cardinals
cites work: Identity crises and strong compactness. II: Strong cardinals
cites work: The least measurable can be strongly compact and indestructible
cites work: Indestructible weakly compact cardinals and the necessity of supercompactness for certain proof schemata
cites work: Squares, scales and stationary reflection
cites work: Changing cofinalities and the nonstationary ideal
cites work: On certain indestructibility of strong cardinals and a question of Hajnal
cites work: Extensions with the approximation and cover properties have no new large cardinals
cites work: Gap forcing
cites work: Gap Forcing: Generalizing the Lévy-Solovay Theorem
cites work: The lottery preparation
cites work: Small forcing creates neither strong nor Woodin cardinals
cites work: Q4315890
cites work: Making the supercompactness of \(\nu\) indestructible under \(\nu\)-directed closed forcing
cites work: Measurable cardinals and the continuum hypothesis
cites work: Strong axioms of infinity and elementary embeddings
DOI: 10.1016/J.APAL.2006.05.002
description: scientific article; zbMATH DE number 5121367