Duality between Logics and Equivalence Relations

Assuming w is the only measurable cardinal, we prove: (i) Let ~ be an equivalence relation such that ~ = =z. for some logic L < L* satisfying Robinson's consistency theorem (with L* arbitrary); then there exists a strongest logic L+ « L* such that ~ = =/.+ ; in addition, L+ is countably compact if ~^= . (ii) Let ~ be an equivalence relation such that ~ = =/." for some logic L° satisfying Robinson's consistency theorem and whose sentences of any type t are (up to equivalence) equinumerous with some cardinal kt; then L° is the unique logic L such that ~ = =;.; furthermore, L° is compact and obeys Craig's interpolation theorem. We finally give an algebraic characterization of those equivalence relations ~ which are equal to =/. for some compact logic L obeying Craig's interpolation theorem and whose sentences are equinumerous with some cardinal. 0. Introduction. This paper is concerned with the following problem: what can we say about the inverse of the map taking logic L into L-elementary equivalence =¡?. We shall derive invertibility results in case L satisfies Robinson's consistency theorem. In studying the interrelations between logics and equivalence relations on structures, the notion of Robinson's consistency has many applications (see [Mu2]): on one hand, any logic L in which Stc¿(r) is a set satisfies Robinson's consistency theorem iff L is compact and satisfies Craig's interpolation theorem, by a result due to the present author and, independently, to Makowsky and Shelah (see [Mu3] and [MSI]); on the other hand, Robinson's consistency only depends on =¿ rather than on L, and =¿ has a simpler structure: in fact, Robinson's consistency has a very neat algebraic characterization in terms of amalgamation and joint embedding properties (see [Mu3] and [Mu6]); furthermore, one can relativize this notion to equivalence relations on smaller classes of structures: thus, for instance, in [Mul] it is proved that on the class of countable structures of finite type there are just two nonpathological equivalence relations satisfying Robinson's consistency, namely s and = ; in the light of the above-mentioned equivalence "Robinson = Craig + Compactness", this might be also regarded as a partial answer to H. Friedman's fourth problem in [Fr] of finding proper extensions of first-order logic still satisfying compactness and interpolation. Concerning Friedman's third problem, too, the techniques developed in [Mu5] for the study of Robinson's consistency in infinitary logics yield such results as "no logic L strictly between Lxu and Lxa0 obeys Craig's Received by the editors August 4, 1980. 1980 Mathematics Subject Classification. Primary 03C95; Secondary 03C30, 03C75, 03C40, 03C55, 03E55. © 1982 American Mathematical Society 0002-9947/81/0000-102 S/$05.75 111 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use