Definition of "well-ordering" [well-ordering]

• A total order of which every nonempty subset has a least element. (noun)
• Present participle of well-order. (verb)

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• "Zermelo introduces axioms of set theory, explicitly formulates AC and uses it to prove the well-ordering theorem, thereby raising a storm of controversy."
• "Zermelo's original purpose in introducing AC was to establish a central principle of Cantor's set theory, namely, that every set admits a well-ordering and so can also be assigned a cardinal number."
• "The need for a specification of infinite sets is crucial in the discussion of the related well-ordering problem."