Robinson's Consistency Theorem in Soft Model Theory

In a soft model-theoretical context, we investigate the properties of logics satisfying the Robinson consistency theorem; the latter is for many purposes the same as the Craig interpolation theorem together with compactness. Applications are given to H. Friedman's third and fourth problem. Introduction. No extension of first order logic is known which is axiomatizable and/or countably compact, and has some kind of interpolation or definability property such as the Robinson consistency theorem or the Craig interpolation theorem. For the case L > L(QX) (where Qx is the quantifier "there exist uncountably many"), by a result in [Hu] we have, on one hand, failure of Craig interpolation in AL((?i) and, on the other hand, in [Mul] it is proved that no countably compact (resp., no axiomatizable) extension of L{Q{) will satisfy the Robinson consistency theorem. Both these results seem to support the feeling expressed in [Br] about the nonexistence of countably compact extensions of L(QX) satisfying the interpolation property. In the general case when L ^ L(Q\), less is known. The Robinson consistency theorem is a very important soft model-theoretical notion; in a somewhat weaker form it was investigated by Makowsky and Shelah in [MS]; in the present form it is studied in [Mul]-[Mu5] and in [MSI]; in [Mu3], [Mu4] and, independently, in [MSI] such identities are proved as Robinson Consistency = Compactness + Craig Interpolation, and Compactness = JEP (i.e. the Joint Embedding Property of L-elementary embeddings). In this paper we give an exposition of the methods and results about the Robinson consistency theorem in abstract (soft) model theory. We assume familiarity with [Fe2], [Fl], [Ba], [MSS] and with [MS, §6]. In Theorem 3.1 we prove that if in L the Robinson consistency theorem holds, then either L is countably compact, or the theories of L can characterize up to isomorphism all the structures whose cardinality is less than <ow. Notice that no special set-theoretical assumption is involved in the proof of this theorem. Corollary 3.3, improving [MS, 6.11], states that if 2" < 2"" for some n e u, then the Received by the editors December 1, 1979. 1980 Mathematics Subject Classification. Primary 03C95, 03C40, 03C75, 03C80; Secondary 03C55, 03C30, 03E55. © 1981 American Mathematical Society 0002 -9947/81 /0000-001 2/$03.7 S 231 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use