Bilangan alef: Perbedaan antara revisi

 

 

 

* himpunan semua bilangan [[pangkat dua|kuadrat]], himpunan semua bilangan [[Pangkat tiga|kubik]], himpunan semua bilangan [[eksponen|pangkat empat]], ...

* himpunan semua [[bilangan genap]], himpunan semua [[bilangan ganjil]],

 

:{1, 3, 5, 7, 9, ..., 2, 4, 6, 8, 10, ...}

 

 

 

 

'''ω₁''' 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 '''ω₁'''.

 

:<math>2^{\aleph_0}=\aleph_1.</math>

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]].

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]]).

 

 

:<math>\aleph_{0} = \omega</math>

:<math>\aleph_{\alpha+1} = \aleph_{\alpha}^+</math>

 

 

:<math>\aleph_{\lambda} = \bigcup_{\beta < \lambda} \aleph_\beta.</math>

 

 

:<math>\alpha\leq\omega_\alpha.</math>

 

:<math>\omega,\ \omega_\omega,\ \omega_{\omega_\omega},\ \ldots.</math>

 

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.

 

Each [[finite set]] is well-orderable, but does not have an aleph as its cardinality.

 

* [[Bilangan kardinal]]

 

{{reflist}}

 

== Pranala luar ==

* {{MathWorld | urlname=Aleph-0 | title=Aleph-0}}

 

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