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- Ordinal_number abstract "In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated.Ordinals were introduced by Georg Cantor in 1883 to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. He derived them by accident while working on a problem concerning trigonometric series.Two sets S and S' have the same cardinality if there is a bijection between them (i.e. there exists a function f that is both injective and surjective, that is it maps each element x of S to a unique element y = f(x) of S' and each element y of S' comes from exactly one such element x of S).If a partial order < is defined on set S, and a partial order <' is defined on set S' , then the posets (S,<) and (S' ,<') are order isomorphic if there is a bijection f that preserves the ordering. That is, f(a) <' f(b) if and only if a < b. Every well ordered set (S,<) is order isomorphic to the set of ordinals less than one specific ordinal number (the order type of (S,<)) under their natural ordering.The finite ordinals (and the finite cardinals) are the natural numbers: 0, 1, 2, …, since any two total orderings of a finite set are order isomorphic. The least infinite ordinal is ω, which is identified with the cardinal number . However in the transfinite case, beyond ω, ordinals draw a finer distinction than cardinals on account of their order information. Whereas there is only one countably infinite cardinal, namely itself, there are uncountably many countably infinite ordinals, namelyω, ω + 1, ω + 2, …, ω·2, ω·2 + 1, …, ω2, …, ω3, …, ωω, …, ωωω, …, ε0, ….Here addition and multiplication are not commutative: in particular 1 + ω is ω rather than ω + 1 and likewise, 2·ω is ω rather than ω·2. The set of all countable ordinals constitutes the first uncountable ordinal ω1, which is identified with the cardinal (next cardinal after ). Well-ordered cardinals are identified with their initial ordinals, i.e. the smallest ordinal of that cardinality. The cardinality of an ordinal defines a many to one association from ordinals to cardinals.In general, each ordinal α is the order type of the set of ordinals strictly less than the ordinal α itself. This property permits every ordinal to be represented as the set of all ordinals less than it. Ordinals may be categorized as: zero, successor ordinals, and limit ordinals (of various cofinalities). Given a class of ordinals, one can identify the α-th member of that class, i.e. one can index (count) them. Such a class is closed and unbounded if its indexing function is continuous and never stops. The Cantor normal form uniquely represents each ordinal as a finite sum of ordinal powers of ω. However, this cannot form the basis of a universal ordinal notation due to such self-referential representations as ε0 = ωε0. Larger and larger ordinals can be defined, but they become more and more difficult to describe. Any ordinal number can be made into a topological space by endowing it with the order topology; this topology is discrete if and only if the ordinal is a countable cardinal, i.e. at most ω. A subset of ω + 1 is open in the order topology if and only if either it is cofinite or it does not contain ω as an element.".
- Ordinal_number thumbnail Omega-exp-omega-labeled.svg?width=300.
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- Ordinal_number wikiPageExternalLink 16.pdf.
- Ordinal_number wikiPageExternalLink index.html.
- Ordinal_number wikiPageExternalLink ordinals.htm.
- Ordinal_number wikiPageExternalLink 117770262.
- Ordinal_number wikiPageExternalLink catalog.php?isbn=9780674324497.
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- Ordinal_number hasPhotoCollection Ordinal_number.
- Ordinal_number id "p/o070180".
- Ordinal_number title "Ordinal Number".
- Ordinal_number title "Ordinal number".
- Ordinal_number urlname "OrdinalNumber".
- Ordinal_number subject Category:Ordinal_numbers.
- Ordinal_number subject Category:Wellfoundedness.
- Ordinal_number comment "In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated.Ordinals were introduced by Georg Cantor in 1883 to accommodate infinite sequences and to classify sets with certain kinds of order structures on them.".
- Ordinal_number label "Liczby porządkowe".
- Ordinal_number label "Nombre ordinal".
- Ordinal_number label "Numero ordinale (teoria degli insiemi)".
- Ordinal_number label "Número ordinal (teoría de conjuntos)".
- Ordinal_number label "Número ordinal".
- Ordinal_number label "Ordinaalgetal".
- Ordinal_number label "Ordinal number".
- Ordinal_number label "Ordinalzahl".
- Ordinal_number label "Порядковое число".
- Ordinal_number label "عدد ترتيبي".
- Ordinal_number label "序数".
- Ordinal_number label "順序数".
- Ordinal_number sameAs Ordinální_číslo.
- Ordinal_number sameAs Ordinalzahl.
- Ordinal_number sameAs Número_ordinal_(teoría_de_conjuntos).
- Ordinal_number sameAs Zenbaki_ordinal.
- Ordinal_number sameAs Nombre_ordinal.
- Ordinal_number sameAs Numero_ordinale_(teoria_degli_insiemi).
- Ordinal_number sameAs 順序数.
- Ordinal_number sameAs 서수.
- Ordinal_number sameAs Ordinaalgetal.
- Ordinal_number sameAs Liczby_porządkowe.
- Ordinal_number sameAs Número_ordinal.
- Ordinal_number sameAs m.05lsj.
- Ordinal_number sameAs Q191780.
- Ordinal_number sameAs Q191780.
- Ordinal_number wasDerivedFrom Ordinal_number?oldid=603665354.
- Ordinal_number depiction Omega-exp-omega-labeled.svg.
- Ordinal_number isPrimaryTopicOf Ordinal_number.
