For Γ a revursively enumerable set of formulae, a structure U on a recursive universe is said to be “Γ-recursively enumerable” if the satisfaction predicate restricted to Γ is recursively enumerable (equivalently, if the formulae of Γ uniformulae of Γ uniformly denote recursively enumerable relations on U).
For recursively enumerable sets Γ1 ⊆ Γ2 of formulae we shall, under certain conditions, characterize structures U with the following properties.
1) Every isomorphism form U to a Γ1-recursively enumerable structure is a recursive isomorphism.
2) Every Γ1-recursively enumerable structure isomorphic to U is recursively isomorphic to U.
3) Every Γ1-recursively enumerable structure isomorphic to U is Γ2-recursively enumerable.