An Extension of Trakhtenbrot's Theorem

The celebrated Trakhtenbrot's theorem states that the set of finitely valid sentences of first-order logic is not computably enumerable. In this note we will extend this theorem by proving that the finite satisfiability problem of any fragment of first-order logic is RE-complete, as long as it has an effective syntax, it is equi-expressive with first-order logic over finite models and it is effectively closed under conjunction.