  In 1950, B.A. Trakhtenbrot showed that the set of first-order tautologies
associated to finite models is not recursively enumerable. In 1999, P. H\'ajek
generalized this result to the first-order versions of \L ukasiewicz, G\"odel
and Product logics. In this paper we extend the analysis to the first-order
axiomatic extensions of MTL. Our main result is the following. Let L be an
axiomatic extension L of MTL s.t. TAUT$_\text{L}$ is decidable, and whose
corresponding variety is generated by a chain: for every generic L-chain
$\mathcal{A}$ the set fTAUT$^\mathcal{A}_{\forall}$ (the set of first-order
tautologies associated to the finite $\mathcal{A}$-models) is $\Pi_1$.
Moreover, if in addition L is an extension of BL or an extension of SMTL or an
extension of WNM, then for every generic L-chain $\mathcal{A}$ the set
fTAUT$^\mathcal{A}_{\forall}$ is $\Pi_1$-complete. More in general, for every
axiomatic extension L of MTL s.t. TAUT$_\text{L}$ is decidable there is no
L-chain $\mathcal{A}$ such that L$\forall$ is complete w.r.t. the class of
finite $\mathcal{A}$-models. We have negative results also if we expand the
language with the $\Delta$ operator.
