Classification theory for non-elementary classes. I: The number of uncountable models of \(\psi \in L_{\omega _ 1,\omega}\)
type: Item
instance of: scholarly article
title: Classification theory for non-elementary classes. I: The number of uncountable models of \(\psi \in L_{\omega _ 1,\omega}\)
zbMATH Open document ID: 0552.03019
DOI: 10.1007/BF02761954
author: S. Shelah
published in: Israel Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: The paper is related to the following problem due to Baldwin: Can a sentence of the language L(Q) have exactly one uncountable model? The author continues his investigation from one of his previous papers [Isr. J. Math. 20, 127-148 (1975; Zbl 0324.02038)] in the context of \(L_{\omega_ 1,\omega}\). Assuming that \(2^{\aleph_ n}<2^{\aleph_{n+1}}\) holds for every \(n<\omega\), the main theorem states that every sentence in \(L_{\omega_ 1,\omega}\) has many non- isomorphic models of size \(\aleph_ n\) for some \(n<\omega\) or has models in all infinite cardinalities. It follows from the theorem that every sentence in \(L_{\omega_ 1,\omega}\) has at least \(2^{\aleph_ 1}\) non-isomorphic uncountable models. The central idea of the paper is to transfer properties of models of cardinality \(\aleph_ n\) to countable models and then back from countable models to models of higher cardinalities. To this end, the author introduces the concepts of n-dimensional diagrams, good sets and generalized amalgamation which culminate in the concept of excellent classes - the cutting edge for classification of atomic models of some countable theories. Excellent classes are parallel to totally transcendental theories, they have models in all infinite cardinalities, satisfy the amalgamation property and Łos conjecture. A non-structure theorem is proved for non-excellent classes stating that they have many non-isomorphic models in some \(\aleph_ m\) if they have an uncountable model and if \(2^{\aleph_ n}<2^{\aleph_{n+1}}\) holds for an appropriate n.
Mathematics Subject Classification ID: 03C52
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 03C70
zbMATH DE Number: 3878903
zbMATH Keywords: infinitary languages
zbMATH Keywords: non-elementary classes
zbMATH Keywords: number of models
zbMATH Keywords: uncountable model
zbMATH Keywords: classification of atomic models
zbMATH Keywords: countable theories
zbMATH Keywords: totally transcendental theories
reviewed by: Olga Stepankova
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02761954
OpenAlex ID: W1965265111
description: scientific article; zbMATH DE number 3878903