Changing cofinalities and collapsing cardinals in models of set theory
type: Item
instance of: scholarly article
title: Changing cofinalities and collapsing cardinals in models of set theory
zbMATH Open document ID: 1018.03041
DOI: 10.1016/S0168-0072(02)00067-2
author: Miloš S. Kurilić
published in: Annals of Pure and Applied Logic
publication date: 2003-03-16T00:00:00Z
review text: The author studies the effect of extending a model \(M\) of ZFC to a model \(N\) of ZFC where the regular cardinal \(\kappa_{1}\) of \(M\) is collapsed to the cardinal \(\kappa_{0}\) in \(N\) and its new cofinality \(\rho\) is \(<\kappa_{0}\), that is \(N\vDash \text{cf}(\kappa_{1})=\rho<\kappa_{0}=|\kappa_{1}|<\kappa_{1}\). He shows that if \((^{\rho}\kappa_{0})^{N}=(^{\rho}\kappa_{0})^{M}\), then for each \(M\)-cardinal \(\lambda\) such that \(M\vDash \kappa_{1}<\lambda< \text{cc}(P(\kappa_{1})/[\kappa_{1}]^{<\kappa_{1}})\) it is the case that \(|\lambda|^{N}\leq |(\kappa_{0}^{\rho})^{M}|^{N}=(\kappa_{0}^{\rho})^{N}\). Moreover, if \((\kappa_{0}^{\rho})^{N}=\kappa_{0}\) then \(|\lambda|^{N}=\kappa_{0}\). For \(N=M[f]\), where \(f\colon\rho\rightarrow\kappa_{1}\) is an unbounded mapping, \(N\) is a \(|\lambda|=\kappa_{0}\)-minimal extension. Applications are made to the forcing notions of Bukovský and Namba.
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E40
Mathematics Subject Classification ID: 03C62
zbMATH DE Number: 1882257
zbMATH Keywords: forcing
zbMATH Keywords: models of ZFC
zbMATH Keywords: minimal models
zbMATH Keywords: changing cofinalities
zbMATH Keywords: collapsing cardinals
reviewed by: J. M. Plotkin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1882257