Formally, the most inclusive definition of an enumeration of a set "S" is any surjection from an arbitrary index set "I" onto "S". In this broad context, every set "S" can be trivially enumerated by the identity function from "S" onto itself. If one does "not" assume the axiom of choice or one of its variants, "S" need not have any well-ordering. Even if one does assume the axiom of choice, "S" need not have any natural well-ordering. This general definition therefore lends itself to a counting notion where we are interested in "how many" rather than "in what order." In practice, this broad meaning of enumeration is often used to compare the relative sizes or cardinalities of different sets. If one works in Zermelo–Fraenkel set theory without the axiom of choice, one may want to impose the additional restriction that an enumeration must also be injective (without repetition) since in this theory, the existence of a surjection from "I" onto "S" need not imply the existence of an injection from "S" into "I". In computability theory one often considers countable enumerations with the added requirement that the mapping from formula_1 (set of all natural numbers) to the enumerated set must be computable. The set being enumerated is then called recursively enumerable (or computably enumerable in more contemporary language), referring to the use of recursion theory in formalizations of what it means for the map to be computable. In this sense, a subset of the natural numbers is computably enumerable if it is the range of a computable function. In this context, enumerable may be used to mean computably enumerable. However, these definitions characterize distinct classes since there are uncountably many subsets of the natural numbers that can be enumerated by an arbitrary function with domain ω and only countably many computable functions. A specific example of a set with an enumeration but not a computable enumeration is the complement of the halting set. Furthermore, this characterization illustrates a place where the ordering of the listing is important. There exists a computable enumeration of the halting set, but not one that lists the elements in an increasing ordering. If there were one, then the halting set would be decidable, which is provably false. In general, being recursively enumerable is a weaker condition than being a decidable set.