Gödel logics: Prenex fragments

In this paper, we provide a complete classification for the first-order G\"odel logics concerning the property that the formulas admit logically equivalent prenex normal forms. We show that the only first-order G\"odel logics that admit such prenex forms are those with finite truth value sets since they allow all quantifier-shift rules and the logic \(G_\uparrow\) with only one accumulation point at $1$ in the infinite truth values set. In all the other cases, there are generally no logically equivalent prenex normal forms. We will also see that \(G_\uparrow\) is the intersection of all finite first-order G\"odel logics.\\ The second part of this paper investigates the existence of effective equivalence between the validity of a formula and the validity of some prenex normal form. The existence of such a normal form is obvious for finite valued G\"odel logic and \(G_\uparrow\). G\"odel logics with an uncountable truth value set admit the prenex normal forms if and only if every surrounding of \(0\) is uncountable or \(0\) is an isolated point. Otherwise, uncountable G\"odel logics are not recursively enumerable, however, the prenex fragment is always recursively enumerable. Therefore, there is no effective translation between the valid formula and the valid prenex normal form. However, the existence of effectively constructible validity equivalent prenex forms for the countable case is still up for debate.