Classification theory over a predicate. I
type: Item
instance of: scholarly article
title: Classification theory over a predicate. I
zbMATH Open document ID: 0604.03011
author: S. Shelah
author: Anand Pillay
published in: Notre Dame Journal of Formal Logic
publication date: 1985-01-01T00:00:00Z
review text: If T is a complete first-order theory and P a unary predicate in its language, the authors study the structre and number of models M of T over the substructure \(M^ P\) of M determined by the universe \(P^ M\). It is shown that, given N, those models of T with \(M^ P=N\) can be coded to some extent by L'-reducts of expansions of N to a certain theory T'', where \(L(T)\subseteq L'\subseteq L(T'')\). The number of such reducts is estimated. This generalizes the context of the Chang-Makkai theorem. It is shown that if for some N of power \(\Lambda\), the number of non-N- isomorphic models M of T, with \(M^ P=N\), \(| M| =\Lambda\) is not too big, then the set of types \(tp(\bar a/P^ M)\) is definable over \(P^ M\) for every model M of T and every \(\bar a\in M\). Assuming the above definability condition and \(T=T''\) some non-structure theorems are proved.
Mathematics Subject Classification ID: 03C45
zbMATH DE Number: 3974950
zbMATH Keywords: non-structure theorems
reviewed by: Olga Stepankova
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1093870929
OpenAlex ID: W2081511772
DOI: 10.1305/NDJFL/1093870929
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description: scientific article; zbMATH DE number 3974950