An Elementary Definability Theorem for First Order Logic

In this paper, we will present a definability theorem for first order logic. This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language , then, clearly, any definable subset S ⊂ M (i.e., a subset S = { a ∣ M ⊨ φ( a )} defined by some formula φ) is invariant under all automorphisms of M . The same is of course true for subsets of M n defined 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 definable by a formula . Our presentation is entirely selfcontained, and only requires familiarity with the most elementary properties of model theory. In particular, we have added a first section in which we review the basic definitions concerning Boolean valued models.