  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.
