Categoricity, amalgamation, and tameness
type: Item
instance of: scholarly article
title: Categoricity, amalgamation, and tameness
zbMATH Open document ID: 1181.03038
author: N. E. Zubov
published in: Israel Journal of Mathematics
publication date: 2009-09-03T00:00:00Z
review text: The categoricity problem in \(L_{\omega_1, \omega}\) is considered. For every \(k\) with \(2 \leq k < \omega\), an \(L_{\omega_1, \omega}\)-sentence \(\varphi_k\) is built satisfying the following conditions: (1) \(\varphi_k\) is categorical in \(\mu\) when \(\mu \leq \aleph_{k-2}\) but not when \(\mu > \aleph_{k-2}\); (2) \(\varphi_k\) is not Galois stable in \(\aleph_{k-2}\); (3) \(\varphi_k\) has the disjoint amalgamation property. Moreover, for \(k > 2\), (4) \(\varphi_k\) is \((\aleph_0, \, \aleph_{k-3})\)-tame, and indeed first-order types determine Galois types over models of cardinality \(\leq \aleph_{k-3}\); (5) \(\varphi_k\) is Galois-stable in \(\aleph_m\) for \(m \leq k-3\); (6) \(\varphi_k\) is not \((\aleph_{k-3}, \, \aleph_{k-2})\)-tame. Also, it is shown that models of \(\varphi_k\) are a model-complete abstract elementary class AEC. The construction refers to earlier work of Hart and Shelah, but adds new significant contributions with respect to amalgamation, tameness and stability. In particular it provides a new example of non-tame AEC (simpler than that previously given by the first author and Shelah).
reviewed by: Carlo Toffalori
Mathematics Subject Classification ID: 03C35
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 03C48
Mathematics Subject Classification ID: 03C75
zbMATH DE Number: 5601722
zbMATH Keywords: categoricity
zbMATH Keywords: amalgamation
zbMATH Keywords: tameness
zbMATH Keywords: Galois type
zbMATH Keywords: abstract elementary class
MaRDI profile type: Publication
full work available at URL: s11856-009-0035-8
OpenAlex ID: W2000276324
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DOI: 10.1007/S11856-009-0035-8
description: scientific article; zbMATH DE number 5601722