Bilangan alef: Perbedaan antara revisi

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[[ImageBerkas:Aleph0.svg|thumb|right|150px|Alef-nol (''aleph-null''), [[bilangan kardinal]] [[tak hingga|tak terhingga]] terkecil]]
'''Bilangan alef''' ({{lang-en|aleph number}}) dalam [[teori himpunan]] (suatu bidang [[matematika]]) adalah suatu urutan bilangan yang digunakan untuk melambangkan [[bilangan kardinal|kardinalitas]] (atau ukuran) dari himpunan tak terhingga (''infinite set''). Dinamakan menurut simbol yang dipakai, yaitu [[abjad Ibrani|huruf Ibrani]] "[[alef]]" (<math>\aleph</math>).
 
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The aleph numbers differ from the [[Extended real number line|infinity]] (∞) commonly found in algebra and calculus. Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme [[limit (mathematics)|limit]] of the [[real number line]] (applied to a [[function (mathematics)|function]] or [[sequence (mathematics)|sequence]] that "[[divergent series|diverges]] to infinity" or "increases without bound"), or an extreme point of the [[extended real number line]].
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== Alef-nol ==
<math>\aleph_0</math> adalah kardinalitas dari semua [[bilangan asli]], dan merupakan suatu [[bilangan transfinit|"bilangan transfinit" atau "kardinal tak terhingga"]]. Himpunan semua [[bilangan ordinal]] finit, dinamakan '''ω''' atau '''ω<sub>0</sub>''', mempunyai kardinalitas <math>\aleph_0</math>. Suatu himpunan mempunyai kardinalitas <math>\aleph_0</math> [[jika dan hanya jika]] bilangan itu [[:en:countably infinite|terhitung sebagai tak terhingga]], yaitu, ada [[:en:bijection|bijeksi]] (kesesuaian satu lawan satu) di antaranya dan bilangan-bilangan asli. Contoh-contoh himpunan semacam itu adalah:
 
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If the [[axiom of countable choice]] (a weaker version of the [[axiom of choice]]) holds, then <math>\aleph_0</math> is smaller than any other infinite cardinal.
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== Alef-satu ==
<math>\aleph_1</math> adalah kardinalitas dari himpunan semua [[bilangan ordinal]] yang terhitung, disebut '''ω<sub>1</sub>''' atau (kadang-kadang) '''Ω'''. '''ω<sub>1</sub>''' sendiri adalah suatu bilangan ordinal yang lebih besar dari semua bilangan ordinal yang terhitung, sehingga merupakan suatu [[:en:uncountable set|himpunan tak terhitung]]. Jadi, <math>\aleph_1</math> berbeda dari <math>\aleph_0</math>. Definisi <math>\aleph_1</math> menyiratkan (dalam ZF, [[:en:Zermelo–Fraenkel set theory|teori himpunan Zermelo–Fraenkel]] ''tanpa'' aksioma pilihan) bahwa tidak ada bilangan ordinal antara <math>\aleph_0</math> dan <math>\aleph_1</math>.<!-- If the [[axiom of choice]] (AC) is used, it can be further proved that the class of cardinal numbers is [[totally ordered]], and thus <math>\aleph_1</math> is the second-smallest infinite cardinal number. Using AC we can show one of the most useful properties of the set '''ω<sub>1</sub>''': any countable subset of '''ω<sub>1</sub>''' has an upper bound in '''ω<sub>1</sub>'''. (This follows from the fact that a countable union of countable sets is countable, one of the most common applications of AC.) This fact is analogous to the situation in <math>\aleph_0</math>: every finite set of natural numbers has a maximum which is also a natural number, and [[Union (set theory)#Finite unions|finite unions]] of finite sets are finite.
 
'''ω<sub>1</sub>''' is actually a useful concept, if somewhat exotic-sounding. An example application is "closing" with respect to countable operations; e.g., trying to explicitly describe the [[sigma-algebra|σ-algebra]] generated by an arbitrary collection of subsets (see e.&nbsp;g. [[Borel hierarchy]]). This is harder than most explicit descriptions of "generation" in algebra ([[vector space]]s, [[group theory|group]]s, etc.) because in those cases we only have to close with respect to finite operations—sums, products, and the like. The process involves defining, for each countable ordinal, via [[transfinite induction]], a set by "throwing in" all possible countable unions and complements, and taking the union of all that over all of '''ω<sub>1</sub>'''.
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== Hipotesis continuum ==
{{main|Continuum hypothesis}}
{{see also|bilangan beth}}
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CH is independent of ZFC: it can be neither proven nor disproven within the context of that axiom system (provided that ZFC is consistent). That it is consistent with ZFC was demonstrated by [[Kurt Gödel]] in 1940 when he showed that its negation is not a theorem of ZFC. That it is independent of ZFC was demonstrated by [[Paul Cohen (mathematician)|Paul Cohen]] in 1963 when he showed, conversely, that the CH itself is not a theorem of ZFC by the (then novel) method of [[Forcing (mathematics)|forcing]].
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== Alef-ω ==
Secara konvensional, bilangan ordinal tak terhingga terkecil dilambangkan dengan ω, dan bilangan kardinal <math>\aleph_\omega</math> merupakan batas atas terkecil dari
:<math>\left\{\,\aleph_n : n\in\left\{\,0,1,2,\dots\,\right\}\,\right\}</math>
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Aleph-ω is the first uncountable cardinal number that can be demonstrated within Zermelo–Fraenkel set theory ''not'' to be equal to the cardinality of the set of all [[real number]]s; for any positive integer n we can consistently assume that <math>2^{\aleph_0} = \aleph_n</math>, and moreover it is possible to assume <math>2^{\aleph_0}</math> is as large as we like. We are only forced to avoid setting it to certain special cardinals with [[cofinality]] <math>\aleph_0</math>, meaning there is an unbounded function from <math>\aleph_0</math> to it (see [[Easton's theorem]]).
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== Alef-α untuk α umum ==
Untuk mendefinisikan <math>\aleph_\alpha</math> bagi bilangan ordinal sembarang <math>\alpha</math>, perlu didefinisikan [[:en:successor cardinal|operasi kardinal penerus]], yang diberikan pada setiap bilangan kardinal ρ bilangan kardinal ρ{{sup|+}} berikutnya yang lebih besar dalam [[:en:well-order|urutan teratur]] (jika [[:en:axiom of choice|aksioma pilihan]] masih dipertahankan, inilah bilangan kardinal lebih besar berikutnya).
 
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Any [[inaccessible cardinal|weakly inaccessible cardinal]] is also a fixed point of the aleph function.<ref name="Harris 2009">{{cite web | url=http://kaharris.org/teaching/582/Lectures/lec31.pdf | title=Math 582 Intro to Set Theory, Lecture 31 | publisher=Department of Mathematics, University of Michigan | date=April 6, 2009 | accessdate=September 1, 2012 | author=Harris, Kenneth}}</ref> This can be shown in ZFC as follows. Suppose <math>\kappa = \aleph_\lambda</math> is a weakly inaccessible cardinal. If <math>\lambda</math> were a [[successor ordinal]], then <math>\aleph_\lambda</math> would be a [[successor cardinal]] and hence not weakly inaccessible. If <math>\lambda</math> were a [[limit ordinal]] less than <math> \kappa </math>, then its [[cofinality]] (and thus the cofinality of <math>\aleph_\lambda</math>) would be less than <math>\kappa </math> and so <math>\kappa </math> would not be regular and thus not weakly inaccessible. Thus <math>\lambda \geq \kappa </math> and consequently <math>\lambda = \kappa </math> which makes it a fixed point.
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== Peranan aksioma pilihan ==
 
Kardinalitas suatu [[bilangan ordinal]] tak terhingga adalah sebuah bilangan alef. Setiap bilangan alef adalah kardinalitas sejumlah bilangan ordinal. Yang terkecil di antaranya adalah [[:en:initial ordinal|ordinal awal]]nya. Setiap himpunan yang kardinalitasnya adalah suatu bilangan alef adalah [[:en:equinumerous|ekuinumeral]] dengan suatu bilangan ordinal dan karenanya dapat tertata baik (''well-orderable'').
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* [[Bilangan kardinal]]
 
== Referensi ==
{{reflist}}
 
== Pranala luar ==
* {{springer|title=Aleph-zero|id=p/a011280}}
* {{MathWorld | urlname=Aleph-0 | title=Aleph-0}}
 
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{{DEFAULTSORT:Bilangan alef}}
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