The theorems of Beth and Craig in abstract model theory. III: \(\Delta\)- logics and infinitary logics
type: Item
instance of: scholarly article
title: The theorems of Beth and Craig in abstract model theory. III: \(\Delta\)- logics and infinitary logics
zbMATH Open document ID: 0698.03029
DOI: 10.1007/BF02937304
author: S. Shelah
published in: Israel Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: This paper deals with interpolation and definability properties in abstract model theory. Two papers on the theorems of Beth and Craig for generalized logics (Parts I and II) were previously published by the author jointly with \textit{J. A. Makowsky} [Trans. Am. Math. Soc. 256, 215- 239 (1979; Zbl 0428.03032); Arch. Math. Logik Grundlagenforsch. 21, 13-35 (1981; Zbl 0472.03028)]. In the present paper, a general method is developed yielding counterexamples to Beth's definability and weak definability. One such counterexample is given by the \(\Delta\)-closure of the infinitary logic \(L_{\infty \omega}\). The method is also applied to several cardinality and infinitary logics.
Mathematics Subject Classification ID: 03C95
Mathematics Subject Classification ID: 03C80
Mathematics Subject Classification ID: 03C40
Mathematics Subject Classification ID: 03C75
zbMATH DE Number: 4143965
zbMATH Keywords: generalized quantifiers
zbMATH Keywords: Craig theorem
zbMATH Keywords: Beth theorem
zbMATH Keywords: cardinality logic
zbMATH Keywords: interpolation
zbMATH Keywords: definability
zbMATH Keywords: generalized logics
zbMATH Keywords: infinitary logic
reviewed by: Daniele Mundici
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4143965