First-order Gِdel logics

First-order Gِdel logics are a family of finite- or infinite-valued logics where the sets of truth values V are closed subsets of [ 0 , 1 ] containing both 0 and 1. Different such sets  V in general determine different Gِdel logics  G V (sets of those formulas which evaluate to 1 in every interpretation into  V ). It is shown that G V is axiomatizable iff V is finite, V is uncountable with 0 isolated in V , or every neighborhood of 0 in V is uncountable. Complete axiomatizations for each of these cases are given. The r.e. prenex, negation-free, and existential fragments of all first-order Gِdel logics are also characterized.