The theorems of Beth and Craig in abstract model theory. I. The abstract setting

In the context of abstract model theory various definability properties, their interrelations and their relation to compactness are investigated. Introduction. This is the first of three papers on the analogues of the theorems of Beth and Craig of first order logic in abstract model theory. They grew out of an unpublished preprint [MSf which was revised and extended several times by results of both authors as well as other people. They unify results due to Badger, Ebbinghaus, Friedman, Gostanian, Gregory, Hrbacek, Hutchinson, Kaufmann, Magidor, Malitz, Makkai, Makowsky, Paulos, Shelah and Stavi. In this paper we present the abstract setting; we suppose that the reader is familiar with a standard text on model theory such as [BS] and [CK], with Barwise's [Bal] and [MSS]. The main results here are that: Beth's theorem together with a Feferman-Vaught theorem for tree-like sums implies a weak form of Robinson's consistency lemma (5.4) and the Robinson consistency lemma together with the Feferman-Vaught theorem for pairs implies full compactness (6.2). In [MS2] and [MS3], the continuations of the present paper, we present applications of the general theory to particular logics. [MS2] is devoted to compact logics and some new logics are introduced, and [MS3] is devoted to infinitary logics and A-logics. Some complicated constructions are presented in detail there. There are three aspects of Craig's interpolation theorem and its corollary, Beth's definability theorem: (A) Philosophical. Every implicit definition is equivalent to an explicit definition. Received by the editors July 13, 1976 and, in revised form, March 31, 1978, May 25, 1978 and September 15, 1978. AMS (MOS) subject classifications (1970). Primary 02B20,02B25,02H10,02H13. 'Cf. [Am]. 2This has been circulating since Spring, 1975. © 1979 American Mathematical Society

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