In this paper, we provide a complete classification for the first-order Goedel logics concerning the property that the formulas admit logically equivalent prenex normal forms. We show that the only first-order Goedel 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 value 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 Goedel 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 Goedel logic and $G_\uparrow$. Goedel 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 Goedel 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中公式可允许逻辑等价前束范式的性质进行了完整分类。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）以及具有无限真值集且仅在1处存在唯一聚点的逻辑$G_\uparrow$。在所有其他情况下，通常不存在逻辑等价的前束范式。我们还将证明$G_\uparrow$是所有有限一阶哥德尔逻辑的交集。本文第二部分探讨了公式有效性与某前束范式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$G_\uparrow$，这种等价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集或0是孤立点。否则，不可数哥德尔逻辑不是递归可枚举的，然而其前束片段总是递归可枚举的。因此，有效公式与有效前束范式之间不存在有效转换关系。但对于可数情形，是否存在可有效构造的有效等价前束范式仍有待商榷。