Mathematical Logic Xavier

In any model theoretic logic, Beth’s definability property together with Feferman-Vaught’s uniform reduction property for pairs imply recursive compactness, and the existence of models with infinitely many automorphisms for sentences having infinite models. The stronger Craig’s interpolation property plus the uniform reduction property for pairs yield a recursive version of Ehrenfeucht-Mostowski’s theorem. Adding compactness, we obtain the full version of this theorem. Various combinations of definability and uniform reduction relative to other logics yield corresponding results on the existence of non-rigid models. The celebrated theorem of Ehrenfeucht and Mostowski states that any first order theory having infinite models must have non rigid models. In its strongest form, it claims that any linearly ordered set may be embedded as a set of indiscernibles in a model of the theory, so that all the automorphisms of the order extend to automorphisms of the model (cf. [EM], [Ho]). The known proofs of the theorem utilize compactness and Ramsey’s theorem or ultrafilter arguments combining both to get models with the given ordered set as a set of indiscernibles, and they use the fact that first order logic is generated by the existential quantifier to Skolemize over the indiscernibles. These techniques have been adapted successfully to obtain analogs of the Ehrenfeucht-Mostowski theorem for the infinitary logics Lκω by Chang [Ch] and others, following Morley [Mo], and for logics with additional quantifiers like L(Qα), L(Q cof ω ) or L(aa) by Ebbinghaus [E1] and Otto [O]. In this paper, we show that the familiar definability properties, together with the Feferman-Vaught uniform reduction property for pairs (URP), imply versions of increasing strength of the Ehrenfeucht-Mostowski theorem for arbitrary model theoretic logics. Thus, Beth’s definability property andURP imply the existence of models with infinite automorphism group for any relativized projective class having infinite or arbitrarily large finite models. Recursive compactness follows (Sec. 2). Under Craig’s interpolation lemma, this may be improved to a recursive version of Ehrenfeucht-Mostowski’s theorem, implying, for example, that any recursive theory with infinite models has models with the ordered rational numbers as indiscernibles (Sec. 3).Adding compactness yields the full Ehrenfeucht-Mostowski theorem except, perhaps, for its functorial character (Sec. 4). Other combinations X. Caicedo: Departamento de Matemáticas, Universidad de los Andes A.A. 4976, Bogotá, Colombia. e-mail: [email protected]