Editing Aleph number (section)

==Fixed points of omega==
For any ordinal <math>\alpha</math> we have <math>\alpha \le \omega_\alpha</math>.

In many cases <math>\omega_\alpha</math> is strictly greater than ''α''. For example, it is true for any successor [[Ordinal number|ordinal]]: <math>\alpha + 1 \le \omega_{\alpha + 1}</math> holds. 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}}, \cdots </math>

which is sometimes denoted <math display="inline">\omega_{\omega_{\ddots}}</math>.

Any [[inaccessible cardinal|weakly inaccessible cardinal]] is also a fixed point of the aleph function.<ref name=Harris-2009-04-06-Math-582>
{{cite web
 | author=Harris, Kenneth A.
 | date=6 April 2009
 | title=Lecture&nbsp;31
 | series=Intro to Set Theory
 | id=Math&nbsp;582 
 | department=Department of Mathematics
 | publisher=[[University of Michigan]]
 | website=kaharris.org
 | url=http://kaharris.org/teaching/582/Lectures/lec31.pdf
 | access-date=September 1, 2012
 | archive-url=https://web.archive.org/web/20160304121941/http://kaharris.org/teaching/582/Lectures/lec31.pdf
 | archive-date=March 4, 2016
 | url-status=dead
 | df=mdy-all
}}
</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 \ge \kappa</math> and consequently <math>\lambda = \kappa</math> which makes it a fixed point.