Erratum to: 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 Lukasiewicz, Gödel and Product logics, w.r.t. their standard algebras. In this talk 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 non-trivial MTL-chains: then the set of all first-order tautologies associated to the finite models over chains in K, fTAUT∀ , is Π 1 -hard. Let TAUTK be the set of propositional tautologies of K: if TAUTK is decidable, we have that fTAUT∀ is in Π 0 1 . We have similar results also if we expand the language with the ∆ operator.