The short extenders gap two forcing is of Prikry type
type: Item
instance of: scholarly article
title: The short extenders gap two forcing is of Prikry type
zbMATH Open document ID: 1191.03037
author: Carmi Merimovich
published in: Archive for Mathematical Logic
publication date: 2009-12-14T00:00:00Z
review text: Suppose \(\kappa = \bigcup_{n<\omega} \kappa_n\), where \(\langle\kappa_n: n<\omega\rangle\) is an increasing sequence of cardinals with the property that for each \(n<\omega\), there is a \(\langle\kappa_n, \kappa_n^{+n+2}\rangle\) extender. Let \((\mathbb P,\leq,\leq^\ast)\) be the associated short extender gap-2 notion of forcing. This forcing is due to \textit{M. Gitik} [Ann. Pure Appl. Logic 80, No.~1, 17--33 (1996; Zbl 0849.03039)], who proved that \(\mathbb P\) preserves cardinals and adds no new bounded subsets of \(\kappa\), and that in \(V^{\mathbb P}\), \(2^\kappa = \kappa^{++}\). The author establishes that \((\mathbb P,\leq, \leq^\ast)\) is of Prikry type, which means that for each condition \(p\in\mathbb P\) and each formula \(\sigma\) in the \(\mathbb P\)-forcing language, there is \(p^\ast \leq^\ast p\) deciding \(\sigma\).
reviewed by: Pierre Matet
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E55
zbMATH DE Number: 5646249
zbMATH Keywords: Singular Cardinal Hypothesis
zbMATH Keywords: Prikry property
zbMATH Keywords: extender-based forcing
MaRDI profile type: Publication
full work available at URL: s00153-009-0147-1
OpenAlex ID: W1968182665
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cites work: Indiscernible sequences for extenders, and the singular cardinal hypothesis
DOI: 10.1007/S00153-009-0147-1
description: scientific article; zbMATH DE number 5646249