Bilangan ordinal: Perbedaan antara revisi

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[[FileBerkas:omega-exp-omega-labeled.svg|thumb|300px|Representasi bilangan ordinal sampai &omega;<sup>&omega;</sup>. Tiap putaran spiral mewakili satu pangkat &omega;]]
'''Bilangan ordinal''' dalam [[teori himpunan]] adalah jenis tatanan dari suatu himpunan yang teratur baik. Biasanya diidentifikasi dengan himpunan transitif hereditari. Bilangan ordinal merupakan perluasan [[bilangan asli]], berbeda dengan [[integer]] dan dengan [[bilangan kardinal]]. Sebagaimana jenis bilangan lain, bilangan ordinal dapat dijumlahkan, dikalikan, dan dipangkatkan.
 
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Contoh:
* Himpunan bilangan ordinal kurang dari 3 adalah 3 = {{mset| 0, 1, 2 }}, bilangan ordinal terkecil tidak kurang dari 3.
* Himpunan bilangan ordinal terhingga adalah tak terhingga, bilangan ordinal tak terhingga terkecil: ω.
* Himpunan bilangan ordinal terhitung adalah tak terhitung, bilangan ordinal tak terhitung terkecil: ω<sub>1</sub>.
 
== Lihat pula ==
* [[Bilangan kardinal]]
* [[Counting]]
* [[Order topology#Ordinal space|Ordinal space]]
 
== Referensi ==
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== Pustaka ==
* [[Georg Cantor|Cantor, G.]], (1897), ''Beitrage zur Begrundung der transfiniten Mengenlehre. II'' (tr.: Contributions to the Founding of the Theory of Transfinite Numbers II), Mathematische Annalen 49, 207-246 [http://www.archive.org/details/117770262 English translation].
* [[John Horton Conway|Conway, J. H.]] and [[Richard K. Guy|Guy, R. K.]] "Cantor's Ordinal Numbers." In ''The Book of Numbers''. New York: Springer-Verlag, pp.&nbsp;266–267 and 274, 1996.
* Dauben, Joseph Warren, (1990), [http://books.google.com/books?id=-cpFeTPJXDIC&lpg=PA100&ots=-OcWWxn5se&dq=%22second%20number%20class%22&pg=PA95#v=onepage&q=&f=false Georg Cantor: his mathematics and philosophy of the infinite]. Chapter 5: The Mathematics of Cantor's ''Grundlagen''. ISBN 0-691-02447-2
* {{Citation | last = Hamilton | first = A. G. | title = Numbers, Sets, and Axioms : the Apparatus of Mathematics | year = 1982 | location = New York | publisher = Cambridge University Press | isbn = 0-521-24509-5}} See Ch. 6, "Ordinal and cardinal numbers"
* [[Akihiro Kanamori|Kanamori, A.]], [http://math.bu.edu/people/aki/16.pdf ''Set Theory from Cantor to Cohen''], to appear in: Andrew Irvine and John H. Woods (editors), ''The Handbook of the Philosophy of Science'', volume 4, Mathematics, Cambridge University Press.
* {{Citation | last1=Levy | first1=A. | title=Basic Set Theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1979}} Reprinted 2002, Dover. ISBN 0-486-42079-5
* {{Citation | last1=Jech | first1=Thomas | title=[[Set Theory]] | publisher= [[Springer-Verlag]]| location=Berlin, New York | series=Springer Monographs in Mathematics | year=2003}}
* Sierpiński, W. (1965). ''Cardinal and Ordinal Numbers'' (2nd ed.). Warszawa: Państwowe Wydawnictwo Naukowe. Also defines ordinal operations in terms of the Cantor Normal Form.
* [[Patrick Suppes|Suppes, P.]] (1960), ''Axiomatic Set Theory'', D.Van Nostrand Company Inc., ISBN 0-486-61630-4
* {{Citation|last=von Neumann|first=Johann|author-link=John von Neumann|year=1923|title=Zur Einführung der trasfiniten Zahlen|journal=Acta litterarum ac scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, Sectio scientiarum mathematicarum|publisher=|pages=199–208|volume=1|url=http://acta.fyx.hu/acta/showCustomerArticle.action?id=4981&dataObjectType=article&returnAction=showCustomerVolume&sessionDataSetId=39716d660ae98d02&style=}}
* {{Citation|last=von Neumann|first=John|author-link=John von Neumann|editor=Jean van Heijenoort|origyear=1923|date=January 2002|title=From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931|chapter=On the introduction of transfinite numbers|edition=3rd|publisher=Harvard University Press|pages=346–354|isbn=0-674-32449-8|url=http://www.hup.harvard.edu/catalog.php?isbn=9780674324497}} - English translation of {{Harvnb|von Neumann|1923}}.
 
== Pranala luar ==
{{Wiktionary|ordinal}}
* {{springer|title=Ordinal number|id=p/o070180}}
* {{MathWorld | urlname=OrdinalNumber| title=Ordinal Number}}
* [http://www.apronus.com/provenmath/ordinals.htm Ordinals at ProvenMath]
* [http://www.springerlink.com/content/n3797702v6422612/ Beitraege zur Begruendung der transfiniten Mengenlehre] Cantor's original paper published in Mathematische Annalen 49(2), 1897
* [http://mtnmath.com/ord/index.html Ordinal calculator] [[GNU General Public License|GPL'd]] free software for computing with ordinals and ordinal notations
 
{{countable ordinals}}
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{{Teori himpunan}}
 
[[CategoryKategori:Bilangan ordinal| ]]
[[CategoryKategori:Bilangan]]