An Application of Invariant Sets to Global Definability

Vaught's "*-transform method" is applied to derive a global definability theorem of M. Makkai from a classical theorem of Lusin. In a recent article [10], R. L. Vaught introduced a construction which connects the invariant descriptive set theory of "logic actions" with the model theory of the infinitary language L In this paper we will use Vaught's construction to give a short derivation of a recent global definability theorem of M. Makkai [8] from a classical theorem of Lusin on countable-to-one continuous functions. Makkai's theorem may be stated as follows. Assume that p is an arbitrary countable similarity type. Let P be a new n-ary relation symbol and let p, = p + P be the corresponding expansion of p. Given a sentence a E L,,,,(pI) and a p-structure W = (A, S), let MQ() = {P c An (by p) l= } THEOREM 1 (MAKKAI). For each sentence a E L,,,,,(pI) the following are equivalent: (i) For every countable p-structure X, M,(%) is countable. (ii) There exists a set b = {pi (vI ... Vn+;k): i E ws} c Li,(p) such that #l V 3Vn+1 ... Vn+ki 'VV ... Vn(P(VI ... Vn) +-+i) ijc Theorem 1 is the infinitary version of the well-known Chang-Makkai Theorem (cf. [3, 5.3.6]). This first-order result is easily derived from the infinitary version using H. J. Keisler's theory [5] of approximations to infinitary formulas-see Remark II below. We will derive Makkai's theorem from the following theorem of Lusin. f: B -+ Y is countable-to-one if the preimage of every point in Y is countable. A topological space is Polish if it is separable and completely metrizable. (1) Let B be a Borel subset of a Polish space X and supposed is a countable-to-one, continuous function on B to a metric space Y. Then there is a collection {Bj: i Es w} of Borel sets such that B = Uicco B, and eachf B is one-one. A variant of (1) is stated-in Kuratowski [6, ?39,VII, Corollary 5]. A proof may be found in Lusin [7]. Since this last reference is somewhat obscure, we have included a sketch of a proof of (1) in Remark IV below. The central proof of this paper was included in the author's Ph. D. dissertation which was written under the supervision of R. L. Vaught. Thanks are also due to John Burgess for a stimulating conversation regarding Proposition 2. Before proceeding with our proof of Theorem 1, we summarize the material from Vaught [10] which we require. Received November 12, 1976. 'Research partially supported by NSF grant MCS74-08550. 9 ? 1979, Association for Symbolic Logic This content downloaded from 18.7.29.240 on Mon, 11 Nov 2013 13:43:39 PM All use subject to JSTOR Terms and Conditions