Editing Enumeration (section)

=== Ordinals ===

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 number|ordinal]]. Under this definition, an enumeration of a set ''S'' is any [[surjection]] from an ordinal &alpha; onto ''S''. The more restrictive version of enumeration mentioned before is the special case where &alpha; is a finite ordinal or the first limit ordinal &omega;. This more generalized version extends the aforementioned definition to encompass [[Transfinite induction|transfinite]] listings.

Under this definition, the [[first uncountable ordinal|first uncountable ordinal <math>\omega_1</math>]] can be enumerated by the identity function on <math>\omega_1</math> 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.

Since [[set theorist]]s work with infinite sets of arbitrarily large [[cardinality|cardinalities]], the default definition among this group of mathematicians of an enumeration of a set tends to be any arbitrary &alpha;-sequence exactly listing all of its elements. Indeed, in Jech's book, which is a common reference for set theorists, an enumeration is defined to be exactly this. Therefore, in order to avoid ambiguity, one may use the term finitely enumerable or [[denumerable]] to denote one of the corresponding types of distinguished countable enumerations.