Under this definition, the first uncountable ordinal formula_10 can be enumerated by the identity function on formula_10 so that these two notions do not coincide. More generally, it is a theorem of ZF that any well-ordered set can be enumerated under this characterization so that it coincides up to relabeling with the generalized listing enumeration. If one also assumes the Axiom of Choice, then all sets can be enumerated so that it coincides up to relabeling with the most general form of enumerations.