1. well-order - Noun
2. well-order - Verb
well-order (plural well-orders)
(set theory, order theory) A total order of some set such that every nonempty subset contains a least element.
well-order (third-person singular simple present well-orders, present participle well-ordering, simple past and past participle well-ordered)
(set theory, order theory, transitive) To impose a well-order on (a set).
The set of positive integers is well-ordered by the relation ≤.
For example, many results about Borel sets are proved by transfinite induction on the ordinal rank of the set; these ranks are already well-ordered, so the axiom of choice is not needed to well-order them. Source: Internet
The above argument uses the axiom of choice in an essential way at the very beginning, in order to well-order the reals. Source: Internet