Enumeration - Enumeration in Set Theory - Enumeration in Countable Vs. Uncountable Context

Enumeration in Countable Vs. Uncountable Context

The most common use of enumeration in set theory occurs in the context where infinite sets are separated into those that are countable and those that are not. In this case, an enumeration is merely an enumeration with domain ω, the ordinal of the natural numbers. This definition can also be stated as follows:

• As a surjective mapping from (the natural numbers) to S (i.e., every element of S is the image of at least one natural number). This definition is especially suitable to questions of computability and elementary set theory.

We may also define it differently when working with finite sets. In this case an enumeration may be defined as follows:

• As a bijective mapping from S to an initial segment of the natural numbers. This definition is especially suitable to combinatorial questions and finite sets; then the initial segment is {1,2,...,n} for some n which is the cardinality of S.

In the first definition it varies whether the mapping is also required to be injective (i.e., every element of S is the image of exactly one natural number), and/or allowed to be partial (i.e., the mapping is defined only for some natural numbers). In some applications (especially those concerned with computability of the set S), these differences are of little importance, because one is concerned only with the mere existence of some enumeration, and an enumeration according to a liberal definition will generally imply that enumerations satisfying stricter requirements also exist.

Enumeration of finite sets obviously requires that either non-injectivity or partiality is accepted, and in contexts where finite sets may appear one or both of these are inevitably present.