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]],

* himpunan semua [[bilangan prima]], himpunan semua [[bilangan komposit]],

 

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.