Editing Enumeration (section)

==== Properties ====

* There exists an enumeration for a set (in this sense) if and only if the set is [[countable]].
* If a set is enumerable it will have an [[uncountable]] infinity of different enumerations, except in the degenerate cases of the empty set or (depending on the precise definition) sets with one element. However, if one requires enumerations to be injective ''and'' allows only a limited form of partiality such that if ''f''(''n'') is defined then ''f''(''m'') must be defined for all ''m''&nbsp;<&nbsp;''n'', then a finite set of ''N'' elements has exactly ''N''! enumerations.
* An enumeration ''e'' of a set ''S'' with domain <math>\mathbb{N}</math> induces a [[well-order]] ≤ on that set defined by ''s'' ≤ ''t'' if and only if <math>\min e^{-1}(s) \leq \min e^{-1}(t)</math>. Although the order may have little to do with the underlying set, it is useful when some order of the set is necessary.