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We improve a result of Hjorth concerning the nature of thin analytic equivalence relations. The key lemma uses a weakly compact cardinal to construct certain nonstandard models, which Hjorth obtained using #'s. The purpose of this note is to improve the following result of Hjorth [2]. Theorem (Hjorth). Suppose that for every real x, x# exists. Let E be an analytic equivalence relation, El in parameter x0. Then either there exists a perfect set of pairwise E-inequivalent reals or every E-equivalence class has a representative in a set-generic extension of L[xo]. Hjorth's proof makes use of his analysis of nonstandard Ehrenfeucht-Mostowski models built from #'s. Instead, we construct the necessary nonstandard models using infinitary model theory, assuming only the existence of weak compacts. Theorem 1. Suppose that for every real x there is a weakly compact cardinal in L[x]. Then the conclusion of Hjorth's Theorem still holds. The main lemma is the following. Lemma 2. Suppose that there is a weakly compact cardinal in L[x], x a real. Then there is a countable nonstandard w-model Mx of ZF such that x E Mx and L(Mx) = (L in the sense of Mx) has an isomorphic copy in a set-generic extension of L[xo], for any real x0. It is not known if the conclusion of Lemma 2 holds in ZFC alone, for arbitrary x (with ZF replaced by an arbitrary finite subtheory). Proof of Theorem 1 from Lemma 2 (as in Hjorth [2]). Suppose that E is an analytic equivalence relation, El in the parameter x0, and choose an x0-recursive tree T on w x w x ww such that xEy T(x, y) has a branch. For each countable ordinal ae we define xEey ( ) rank(T(x, y)) is at least ae; then Ea is Borel in (x0, c) where c is any real coding ae and E is the intersection of the Ea's. We may assume that each Ea is an equivalence relation (see Theorem 1.4 of Hjorth [2]). A theorem of Harrington and Silver says that a l1-equivalence relation has a perfect set of pairwise inequivalent reals or each equivalence class has a representative constructible from the parameter defining the relation. As Ea is Borel in (xo, c) where c is a Received by the editors September 25, 1995 and, in revised form, November 20, 1995. 1991 Mathematics Subject Classification. Primary 04A15, 03C75. Research supported by NSF Contract # 92-05530. (?)1997 American Mathematical Society

Nonstandard models and analytic equivalence relations