The theorems of Beth and Craig in abstract model theory. III: \(\Delta\)- logics and infinitary logics
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.
scientific article; zbMATH DE number 4143965
scientific article