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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). == Ordinal numbers == (詳細はorder isomorphic to a unique ordinal number, called the order type of the well-ordered set. The position of each element within the ordered set is also given by an ordinal number. In the case of a finite set, the basic operation of counting, to find the ordinal number of a particular object, or to find the object with a particular ordinal number, corresponds to assigning ordinal numbers one by one to the objects. The size (number of elements, cardinal number) of a finite set is equal to the order type. Counting in the everyday sense typically starts from one, so it assigns to each object the size of the initial segment with that object as last element. Note that these numbers are one more than the formal ordinal numbers according to the isomorphic order, because these are equal to the number of earlier objects (which corresponds to counting from zero). Thus for finite ''n'', the expression "''n''-th element" of a well-ordered set requires context to know whether this counts from zero or one. In a notation "β-th element" where β can also be an infinite ordinal, it will typically count from zero. For an infinite set the order type determines the cardinality, but not conversely: well-ordered sets of a particular cardinality can have many different order types. For a countably infinite set, the set of possible order types is even uncountable. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Well-order」の詳細全文を読む スポンサード リンク
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