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- Well-order abstract "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. The set S together with the well-order relation is then called a well-ordered set. The hyphen is frequently omitted in contemporary papers, yielding the spellings wellorder, wellordered, and wellordering.Every non-empty well-ordered set has a least element. Every element s of a well-ordered set, except a possible greatest element, has a unique successor (next element), namely the least element of the subset of all elements greater than s. There may be elements besides the least element which have no predecessor (see Natural numbers below for an example). In a well-ordered set S, every subset T which has an upper bound has a least upper bound, namely the least element of the subset of all upper bounds of T in S.If ≤ is a non-strict well-ordering, then < is a strict well-ordering. A relation is a strict well-ordering if and only if it is a well-founded strict total order. The distinction between strict and non-strict well-orders is often ignored since they are easily interconvertible.Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The well-ordering theorem, which is equivalent to the axiom of choice, states that every set can be well-ordered. If a set is well-ordered (or even if it merely admits a wellfounded relation), 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 (for natural numbers).".
- Well-order wikiPageID "33456".
- Well-order wikiPageRevisionID "593218785".
- Well-order hasPhotoCollection Well-order.
- Well-order subject Category:Mathematical_relations.
- Well-order subject Category:Order_theory.
- Well-order subject Category:Ordinal_numbers.
- Well-order subject Category:Wellfoundedness.
- Well-order type Abstraction100002137.
- Well-order type DefiniteQuantity113576101.
- Well-order type Measure100033615.
- Well-order type Number113582013.
- Well-order type OrdinalNumber113597280.
- Well-order type OrdinalNumbers.
- Well-order comment "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. The set S together with the well-order relation is then called a well-ordered set. The hyphen is frequently omitted in contemporary papers, yielding the spellings wellorder, wellordered, and wellordering.Every non-empty well-ordered set has a least element.".
- Well-order label "Buon ordine".
- Well-order label "Conjunto bien ordenado".
- Well-order label "Dobry porządek".
- Well-order label "Ensemble bien ordonné".
- Well-order label "Relação bem-ordenada".
- Well-order label "Welgeordendheid".
- Well-order label "Well-order".
- Well-order label "Wohlordnung".
- Well-order label "Вполне упорядоченное множество".
- Well-order label "整列集合".
- Well-order label "良序关系".
- Well-order sameAs Dobře_uspořádaná_množina.
- Well-order sameAs Wohlordnung.
- Well-order sameAs Conjunto_bien_ordenado.
- Well-order sameAs Ensemble_bien_ordonné.
- Well-order sameAs Buon_ordine.
- Well-order sameAs 整列集合.
- Well-order sameAs 정렬순서.
- Well-order sameAs Welgeordendheid.
- Well-order sameAs Dobry_porządek.
- Well-order sameAs Relação_bem-ordenada.
- Well-order sameAs m.083h_.
- Well-order sameAs Q659746.
- Well-order sameAs Q659746.
- Well-order sameAs Well-order.
- Well-order wasDerivedFrom Well-order?oldid=593218785.
- Well-order isPrimaryTopicOf Well-order.