Instances of dependent choice and the measurability of \(\aleph _{\omega +1}\)
type: Item
instance of: scholarly article
title: Instances of dependent choice and the measurability of \(\aleph _{\omega +1}\)
zbMATH Open document ID: 0842.03038
DOI: 10.1016/0168-0072(94)00039-6
author: Arthur W. Apter
author: Menachem Magidor
published in: Annals of Pure and Applied Logic
publication date: 1996-07-24T00:00:00Z
review text: Strating from cardinals \(\kappa < \lambda\) such that \(\kappa\) is \(\lambda\)-supercompact and \(\lambda\) is measurable, the authors construct a model for the theory ``\(\text{ZF} + \text{DC}_{\aleph_\omega} + \aleph_{\omega + 1}\) is a measurable cardinal''. (Recall that \(\text{DC}_\kappa\) asserts that given a set \(X\) and \(R \subseteq \left(\bigcup_{\alpha \in \kappa} {^\alpha\kappa}\right) \times X\) such that for every \(g \in \bigcup_{\alpha \in \kappa} {^\alpha \kappa}\), there is an \(x\in X\) with \((g,x) \in R\), there is an \(f \in {^\kappa X}\) such that for every \(\alpha \in \kappa\), \((f|\alpha,\;f(\alpha)) \in R)\). The authors point out that \(\text{DC}_{\aleph_{\omega + 1}}\) is inconsistent with the measurability of \(\aleph_{\omega + 1}\), and that by a result of Shelah, if \(\kappa\) is a singular cardinal such that \(\kappa^+\) is measurable and DC\(_\kappa\) holds, then \(\text{cof}(\kappa) = \omega\).
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E55
Mathematics Subject Classification ID: 03E25
zbMATH DE Number: 792507
zbMATH Keywords: dependent choice
zbMATH Keywords: measurable cardinal
reviewed by: Pierre Matet
MaRDI profile type: Publication
full work available at URL: 0168-0072(94)00039-6
OpenAlex ID: W1970365987
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description: scientific article; zbMATH DE number 792507