TITLE: If the infinite cardinals aleph-null, aleph-two, etc. continue indefinitely, is there any meaning in the idea of aleph-aleph-null?
QUESTION [26 upvotes]: If the infinite cardinals aleph-null, aleph-two, etc. continue indefinitely, is there any meaning in the idea of aleph-aleph-null?
Apologies if this isn't a sensible question, I really don't know too much about these infinite cardinals aside from the basics. I did, however, think that the idea of the "aleph-null"th aleph number was interesting enough to base my username on and my own attempts did not prove fruitful, so I was wondering if anyone here could shed some light. Thanks!
For clarity: I'm asking about $\aleph_{\aleph_0}$ . Thanks!
P.S. I was somewhat unsure about the tags for this, sorry if I accidentally placed it in the wrong category.

REPLY [2 votes]: I realize this question has been answered but I would like to provide a way of visualizing $\aleph_\omega$ and $\aleph_{\omega_\omega}$:
Start with $\aleph_0$ and consider the sequence $\aleph_0, \aleph_1,... \aleph_n,...$ for $n \in \omega.$
This is a strictly monotone increasing sequence that has a limit, namely $\aleph_\omega$, which would make $\aleph_\omega$ the smallest uncountable limit cardinal. This number does have applications. Example: if GHC holds and $\aleph_\lambda = \beth_\lambda$ for all $\lambda$, then $|V_{\omega +\omega}|=\beth_\omega=\aleph_{\omega}$. In other words $\aleph_\omega$ would be the cardinality of the von Neumann universe $V_{\omega +\omega}$ which is a universe sufficient for most ordinary mathematics, and is a model of Zermelo set theory.
As to $\aleph_{\omega_\omega}$, consider the sequence $\aleph_\omega, \aleph_{\omega+1},...,\aleph_{\omega^2},...,\aleph_{\omega^\omega},...,\aleph_{\epsilon_0},...$ 
Notice that this also is a strictly monotone increasing sequence which is countable, the supremum of which is a $\aleph_{\omega_1}$. Continuing on like this yields $\aleph_{\omega_1}, \aleph_{\omega_2},...,\aleph_{\omega_n},...$ the limit of which is $\aleph_{\omega_\omega}$