A Deenability Theorem for Rst Order Logic

In this paper, we will present a deenability theorem for rst order logic. This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us rst observe that if M is a model of a theory T in a language L, then, clearly, any deenable subset S M (i.e., a subset S = fa j M j = '(a)g deened by some formula ') is invariant under all automorphisms of M. The same is of course true for subsets of M n deened by formulas with n free variables. Our theorem states that, if one allows Boolean valued models, the converse holds. More precisely , for any theory T we will construct a Boolean valued model M , in which precisely the T{provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is deenable by a formula of L. Our presentation is entirely selfcontained, and only requires familiarity with the most elementary properties of model theory. In particular, we have added a rst section in which we review the basic deenitions concerning Boolean valued models. The Boolean algebra used in the construction of the model will be presented concretely as the algebra of closed and open subsets of a topological space X naturally associated with the theory T. The construction of this space is closely related to the one in 1]. In fact, one of the results in that paper could be interpreted as a deenability theorem for innnitary logic, using topological rather than Boolean valued models. 1 Preliminary deenitions In this section we review the basic deenitions concerning Boolean valued models (see e.g. 2]). Most readers will be familiar with these notions, and they are advised to skip this section. They should note, however, that our Boolean algebras are not necessarily complete, and that we treat constants and function symbols as functional relations. Let us x a signature S, consisting of constants, function and relation symbols. For simplicity we assume it is a single sorted signature, although this restriction is by no means essential. Let L denote the associated rst order language L !! (S).