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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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==AlephAlef-onesatu==
<math>\aleph_1</math> isadalah thekardinalitas cardinalitydari ofhimpunan the set of all countablesemua [[bilangan ordinal number]]s yang terhitung, calleddisebut '''ω<sub>1</sub>''' oratau (sometimeskadang-kadang) '''Ω'''. This '''ω<sub>1</sub>''' issendiri itselfadalah ansuatu bilangan ordinal numberyang largerlebih thanbesar alldari countablesemua ones,bilangan soordinal ityang isterhitung, ansehingga merupakan suatu [[:en:uncountable set|himpunan tak terhitung]]. Therefore Jadi, <math>\aleph_1</math> isberbeda distinct fromdari <math>\aleph_0</math>. The definition ofDefinisi <math>\aleph_1</math> impliesmenyiratkan (indalam ZF, [[:en:Zermelo–Fraenkel set theory|teori himpunan Zermelo–Fraenkel]] ''withouttanpa'' theaksioma axiom of choicepilihan) thatbahwa notidak cardinalada numberbilangan isordinal betweenantara <math>\aleph_0</math> anddan <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. 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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==TheHipotesis continuum hypothesis==
{{main|Continuum hypothesis}}
{{see also|Bethbilangan numberbeth}}
The [[cardinalityKardinalitas]] ofsuatu the set ofhimpunan [[bilangan real number]]s ([[:en:cardinality of the continuum|kardinalitas continuum]]) isadalah <math>2^{\aleph_0}</math>. ItTidak cannotdapat beditentukan determined fromdari ZFC ([[:en:Zermelo–Fraenkel set theory|teori himpunan Zermelo-Fraenkel]] with thedengan [[:en:axiom of choice|aksioma pilihan]]) wheredi thismana numberbilangan fitsini exactlytepat inmasuk thedalam alephhierarki numberbilangan hierarchyalef, buttetapi it follows frommenuruti ZFC thatbahwa thehipotesis continuum (''continuum hypothesis''), '''CH''', isekuivalen equivalentdengan to thepersamaan identityidentitas
:<math>2^{\aleph_0}=\aleph_1.</math>
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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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==AlephAlef-ω==
ConventionallySecara the smallestkonvensional, infinitebilangan ordinal istak denotedterhingga terkecil dilambangkan dengan ω, anddan thebilangan cardinal numberkardinal <math>\aleph_\omega</math> is themerupakan leastbatas upperatas boundterkecil ofdari
:<math>\left\{\,\aleph_n : n\in\left\{\,0,1,2,\dots\,\right\}\,\right\}</math>
di antara bilangan-bilangan alef.
among alephs.
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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).
Maka bilangan-bilangan alef dapat didefinikan sebagai berikut:
==Aleph-α for general α==
To define <math>\aleph_\alpha</math> for arbitrary ordinal number <math>\alpha</math>, we must define the [[successor cardinal|successor cardinal operation]], which assigns to any cardinal number ρ the next larger [[well-order]]ed cardinal ρ{{sup|+}} (if the [[axiom of choice]] holds, this is the next larger cardinal).
We can then define the aleph numbers as follows:
:<math>\aleph_{0} = \omega</math>
:<math>\aleph_{\alpha+1} = \aleph_{\alpha}^+</math>
anddan foruntuk λ, ansuatu infiniteordinal limit ordinaltak terhingga,
:<math>\aleph_{\lambda} = \bigcup_{\beta < \lambda} \aleph_\beta.</math>
TheOrdinal α-thawal infinitetak initialterhingga ordinalke-α is writtenditulis <math>\omega_\alpha</math>. Kardinalitasnya Its cardinality is writtenditulis <math>\aleph_\alpha</math>. SeeLihat [[:en:initial ordinal|ordinal awal]].
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InDalam ZFC, thefungsi <math>\aleph</math> functionadalah is asuatu [[:en:bijection|bijeksi]] betweenantara thebilangan-bilangan ordinalsordinal anddan thekardinal infinitetak cardinalsterhingga.<ref>{{PlanetMath | urlname=AlephNumbers | title=aleph numbers | id=5710}}</ref>
==Fixed pointsTitik oftetap omega==
ForUntuk anysetiap ordinal α we haveada
:<math>\alpha\leq\omega_\alpha.</math>
InDalam manybanyak caseskasus <math>\omega_{\alpha}</math> issecara strictlysempit greaterlebih thanbesar dari α. For exampleContohnya, forini anybenar successoruntuk setiap ordinal αpenerus this holdsα. <!--There are, however, some limit ordinals which are [[fixed point (mathematics)|fixed point]]s of the omega function, because of the [[fixed-point lemma for normal functions]]. The first such is the limit of the sequence
:<math>\omega,\ \omega_\omega,\ \omega_{\omega_\omega},\ \ldots.</math>
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