Categoricity of theories in \(L_{\kappa \omega}\), with \(\kappa\) a compact cardinal
type: Item
instance of: scholarly article
title: Categoricity of theories in \(L_{\kappa \omega}\), with \(\kappa\) a compact cardinal
zbMATH Open document ID: 0704.03015
DOI: 10.1016/0168-0072(90)90016-U
author: S. Shelah
author: Michael Makkai
published in: Annals of Pure and Applied Logic
publication date: 1990-01-01T00:00:00Z
review text: In this paper the authors generalize the classical Morley categoricity theorem to a particular kind of infinitary languages, \(L_{\kappa \omega}\) with \(\kappa\) a compact cardinal. One of the main results of the paper is the following Theorem: Let \(\kappa\) be an uncountable strongly compact cardinal, \(T\) a theory in a fragment \(F\) of \(L_{\kappa \omega}\) and \(\kappa '=\max (\kappa,| F|)\). Let \(\lambda\) be a successor cardinal with \(\lambda >((\kappa ')^{<\kappa})^+\). If \(T\) is categorical in \(\lambda\) then \(T\) is categorical in every cardinal greater or equal to \(\min (\lambda,\beth_{(2^{\kappa '})^+})\). This paper should be seen as part of the program of classification theory. In Section 1 the necessary preliminaries are given, amalgamation and joint embedding property are investigated. In Section 2 types are studied under the assumption of amalgamation and joint embedding property. In Section 3 the authors deal with order indiscernibles and introduce the Skolem hull. In Section 4 an extension of an elementary part of stability is given. A suitable notion of non-forking over models is developed and used to obtain the categoricity result. Section 5 gives a summary. An appendix contains set-theoretic material used in Section 1. It gives the necessary background concerning modified square-systems.
reviewed by: Martin Weese
Mathematics Subject Classification ID: 03C35
Mathematics Subject Classification ID: 03C75
Mathematics Subject Classification ID: 03C45
zbMATH DE Number: 4154417
zbMATH Keywords: categoricity
zbMATH Keywords: infinitary languages
zbMATH Keywords: compact cardinal
zbMATH Keywords: classification theory
zbMATH Keywords: order indiscernibles
zbMATH Keywords: Skolem hull
zbMATH Keywords: non-forking over models
zbMATH Keywords: modified square-systems
MaRDI profile type: Publication
cites work: Model theory
cites work: Large infinitary languages. Model theory
cites work: A weak version of \(\lozenge\) which follows from \(2^{\aleph_0}<2^{\aleph_1}\)
cites work: Q3852172
cites work: Homogeneous Universal Relational Systems.
cites work: Categoricity in Power
cites work: Q5554949
cites work: Homogeneous Universal Models.
cites work: Classification theory and the number of non-isomorphic models
cites work: Classification theory for non-elementary classes. I: The number of uncountable models of \(\psi \in L_{\omega _ 1,\omega}\)
cites work: Q3781082
cites work: Q3895456
cites work: Around classification theory of models
cites work: Q3778001
cites work: Q4078034
description: scientific article; zbMATH DE number 4154417