The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set.
The precise requirements for an enumeration (for example, whether the set must be finite, or whether the list is allowed to contain repetitions) depend on the discipline of study and the context of a given problem.
There are flourishing subareas in many branches of mathematics concerned with enumerating in this sense objects of special kinds.
According to this characterization, an ordered enumeration is defined to be a surjection (an onto relationship) with a well-ordered domain.
This definition is natural in the sense that a given well-ordering on the index set provides a unique way to list the next element given a partial enumeration.
A set is finite if it can be enumerated by means of a proper initial segment {1, ..., n} of the natural numbers, in which case, its cardinality is n. The empty set is finite, as it can be enumerated by means of the empty initial segment of the natural numbers.
In set theory, there is a more general notion of an enumeration than the characterization requiring the domain of the listing function to be an initial segment of the Natural numbers where the domain of the enumerating function can assume any ordinal.
This more generalized version extends the aforementioned definition to encompass transfinite listings.
Indeed, in Jech's book, which is a common reference for set theorists, an enumeration is defined to be exactly this.
In practice, this broad meaning of enumeration is often used to compare the relative sizes or cardinalities of different sets.
There exists a computable enumeration of the halting set, but not one that lists the elements in an increasing ordering.