The tree property at the double successor of a singular cardinal with a larger gap
type: Item
instance of: scholarly article
title: The tree property at the double successor of a singular cardinal with a larger gap
zbMATH Open document ID: 1387.03036
author: Sy-David Friedman
author: Šárka Stejskalová
author: Radek Honzík
published in: Annals of Pure and Applied Logic
publication date: 2018-04-06T00:00:00Z
review text: For a regular cardinal \(\kappa,\) the tree property at \(\kappa\) is the assertion that there are no \(\kappa\)-Aronszajn trees. The tree property belongs to a family of combinatorial assertions known as compactness principles and has received special attention in the last few years. In the paper under review, the authors study the tree property at the double successor of a singular strong limit cardinal. The first result in this direction is due to \textit{J. Cummings} and \textit{M. Foreman} [Adv. Math. 133, No. 1, 1--32 (1998; Zbl 0949.03039)], who produced, starting from a supercompact cardinal \(\kappa\) and a weakly compact cardinal above it, a model of ZFC in which \(\kappa\) is a singular strong limit cardinal of countable cofinality and the tree property holds at \(\kappa^{++}.\) They also extended their result for \(\kappa=\aleph_{\omega}\). \textit{S.-D. Friedman} and \textit{A. Halilović} [J. Symb. Log. 76, No. 2, 477--490 (2011; Zbl 1221.03053)] have obtained the same results from the weaker assumption of a cardinal \(\kappa\) which is \(H(\lambda)\)-hypermeasurable for some weakly compact cardinal \(\kappa.\) In all these models, \(2^\kappa = \kappa^{++}\) and it is natural to ask if we can obtain the tree property at the double successor of a singular strong limit cardinal \(\kappa\) with \(2^\kappa > \kappa^{++}\). In this paper, the authors prove such a result. Starting from a supercompact cardinal \(\kappa\) and a weakly compact cardinal \(\lambda\) above it, they build a generic extension in which \(\kappa\) is a singular strong limit cardinal of countable cofinality, \(\lambda=\kappa^{++}\), the tree property holds at \(\kappa^{++}\) and \(2^\kappa=\lambda^+=\kappa^{+++}.\) They also sketch how one can make \(2^\kappa\) arbitrary large. The method of the proof is in general based on the argument of Cummings and Foreman [loc.cit.], but is more involved in details since the authors have to add \(\mu\)-many subsets of \(\kappa\) for some \(\mu > \l…
reviewed by: Mohammad Golshani
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E55
zbMATH DE Number: 6856634
zbMATH Keywords: tree property
zbMATH Keywords: singular strong limit cardinal
zbMATH Keywords: countable cofinality
MaRDI profile type: Publication
full work available at URL: j.apal.2018.02.002
OpenAlex ID: W2791087790
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cites work: The tree property
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DOI: 10.1016/J.APAL.2018.02.002
description: scientific article; zbMATH DE number 6856634