Recursively Enumerable Prime Theories in Infinite-Valued Lukasiewicz Logic are not Uniformly Decidable

In infinite-valued Lukasiewicz logic it is well-known that prime theories do not coincide with maximally consistent (complete) theories. It is said that a theory T is prime if, for every pair of formulas ϕ,ψ either ϕ → ψ ∈ T or ψ → ϕ ∈ T. On the other hand, T is maximally consistent if, whenever ϕ 6 ∈ T, some finite Lukasiewicz sum ¬ϕ ⊕... ⊕ ¬ϕ is in T. It is very easy to see that recursively enumerable maximally consistent theories in the infinite-valued Lukasiewicz logic are decidable: simply re-cursively enumerate the theory until ϕ or ¬ϕ⊕...⊕¬ϕ appears. In [MP01] Mundici and Panti proved that the previous result fails for recursively enu-merable prime theories, but that it holds for recursively enumerable prime theories with a finite number of variables. They left open the question whether recursively enumerable prime the-ories with a finite number of variables are uniformly decidable. That is, whether there is a Turing machine that, given a Turing machine whic