In mathematics, a well-order relation (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. Equivalently, a well-ordering is a well-founded total order. The set S together with the well-order relation is then called a well-ordered set. Every element s, except a possible greatest element, has a unique successor (next element), namely the least element of the subset of all elements greater than s. Every subset which has an upper bound has a least upper bound. There may be elements (besides the least element) which have no predecessor. If a set is well-ordered, the proof technique of transfinite induction can be used to prove that a given statement is true for all elements of the set. The observation that the natural numbers are well-ordered by the usual less-than relation is commonly called the well-ordering principle. The well-ordering theorem, which is equivalent to the axiom of choice, states that every set can be well-ordered. The well-ordering theorem is also equivalent to the Kuratowski-Zorn lemma.

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