Successors of singular cardinals and measurability
type: Item
instance of: scholarly article
title: Successors of singular cardinals and measurability
zbMATH Open document ID: 0615.03040
DOI: 10.1016/0001-8708(85)90092-1
author: Arthur W. Apter
published in: Advances in Mathematics
publication date: 1985-01-01T00:00:00Z
review text: In the presence of the axiom of choice measurable cardinals must be very big, but where choice fails, such as in the world of AD, small cardinals such as \(\aleph_ 1\), \(\aleph_ 2\) can be measurable. Earlier relative consistency results of Jech and Takeuti produced inner models where the successor of a regular cardinal can be made measurable. Apter wants to do the same for successors of singular cardinals. Under certain strong hypotheses he succeeds. Theorem 1. Assume \(V\vDash ''ZFC\) \(+\kappa <\lambda\) are cardinals where \(\lambda\) is measurable and \(\kappa\) is \(2^{\lambda}\) supercompact''. Then there is a model of ''ZF \(+\) \(AC_{\kappa}\) \(+\kappa\) is a strong limit cardinal \(+\) \(cof(\kappa)=\omega\) \(+\kappa\) is a Rowbottom cardinal \(+\kappa^+\) is measurable''. The proof uses a generalization of Prikry forcing. Then by a collapsing argument Apter proves (under the same hypotheses as Theorem 1) that there is a model of ''ZF \(+\aleph_{\omega}\) is a Rowbottom cardinal \(+\aleph_{\omega +1}\) is a measurable cardinal''.
Mathematics Subject Classification ID: 03E55
zbMATH DE Number: 3995664
zbMATH Keywords: measurable cardinals
zbMATH Keywords: relative consistency
zbMATH Keywords: inner models
zbMATH Keywords: successors of singular cardinals
zbMATH Keywords: Prikry forcing
reviewed by: J. M. Plotkin
MaRDI profile type: Publication
full work available at URL: 0001-8708(85)90092-1
OpenAlex ID: W2013447938
cites work: An AD-like model
cites work: \(\omega_1\) can be measurable
cites work: Q5596765
cites work: Q4159049
cites work: Infinitary combinatorics and the axiom of determinateness
cites work: Measurable cardinals and the continuum hypothesis
cites work: On the singular cardinals problem. I
cites work: Q5613932
cites work: Strong axioms of infinity and elementary embeddings
cites work: Some strong axioms of infinity incompatible with the axiom of constructibility
cites work: Some applications of model theory in set theory
cites work: A Relativization of Axioms of Strong Infinity to ^|^omega;1
description: scientific article; zbMATH DE number 3995664