Trakhtenbrot Theorem and First-Order Axiomatic Extensions of MTL

In 1950, B.A. Trakhtenbrot showed that the set of first-order tautologies associated to finite models is not recursively enumerable. In 1999, P. Hájek generalized this result to the first-order versions of Łukasiewicz, Gödel and Product logics, w.r.t. their standard algebras. In this paper we extend the analysis to the first-order versions of axiomatic extensions of MTL. Our main result is the following. Let K be a class of MTL-chains. Then the set of all first-order tautologies associated to the finite models over chains in K , {fTAUT^K}_∀ , is {Π^0}_1 -hard. Let TAUT_K be the set of propositional tautologies of K . If TAUT_K is decidable, we have that {fTAUT^K}_∀ is in {Π^0}_1 . We have similar results also if we expand the language with the Δ operator.