Editing Enumeration (section)

=== Comparison of cardinalities ===

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 [[cardinality|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 (mathematics)|injection]] from ''S'' into ''I''.