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.

翻译：本文对一阶哥德尔逻辑进行了完全分类，重点关注公式是否允许逻辑等价的前束范式。我们证明，唯一允许此类前束范式的一阶哥德尔逻辑是那些具有有限真值集的逻辑（因为它们允许所有量词移位规则）以及逻辑 \(G_\uparrow\)（其在无限真值集中仅有一个在 $1$ 处的聚点）。在所有其他情况下，通常不存在逻辑等价的前束范式。我们还将看到，\(G_\uparrow\) 是所有有限一阶哥德尔逻辑的交集。\\ 本文第二部分研究了公式的有效性与某个前束范式的有效性之间是否存在有效等价关系。对于有限值哥德尔逻辑和 \(G_\uparrow\)，这种范式的存在性是显然的。具有不可数真值集的哥德尔逻辑允许前束范式当且仅当 \(0\) 的每个邻域都是不可数的或 \(0\) 是一个孤立点。否则，不可数哥德尔逻辑不是递归可枚举的，然而前束片段总是递归可枚举的。因此，在有效公式与有效前束范式之间不存在有效的转换。然而，对于可数情况，是否存在可有效构造的有效等价前束形式仍有待讨论。