TITLE: Questions about Aleph-Aleph-Null
QUESTION [13 upvotes]: Note: I apologize in advance for not using proper notation on some of these values, but this is literally my first post on this site and I do not know how to display these values correctly.
I recently was looking up facts about different cardinalities of infinity for a book idea, when I found a post made several years ago about $ℵ_{ℵ_0}$
If the infinite cardinals aleph-null, aleph-two, etc. continue indefinitely, is there any meaning in the idea of aleph-aleph-null?
In this post people are talk about the difference between cardinal numbers and how $ℵ_{ℵ_0}$ should instead be $ℵ_ω$. The responses to the post then go on to talk about $ℵ_{ω+1}$, $ℵ_{ω+2}$, and so on.
Anyways, my understanding of the different values of ℵ was that they corresponded to the cardinalities of infinite sets, with $ℵ_0$ being the cardinality of the set of all natural numbers, and that if set X has cardinality of $ℵ_a$, then the cardinality of the powerset of X would be $ℵ_{a+1}$.
With this in mind, I always imagined that if a set Y had cardinality $ℵ_0$, and you found its powerset, and then you found the powerset of that set, and then you found the powerset of THAT set, and repeated the process infinitely you would get a set with cardinality $ℵ_{ℵ_0}$.
So, I guess my question is, in the discussion linked above, when people are talking about $ℵ_{ω+1}$, how is that possible? Because if you take a powerset an infinite number of times, taking one more powerset is still just an infinite number of times, isn't it?
I hope I worded this question in a way that people will understand, and thanks in advance into any insight you can give me about all this.

REPLY [2 votes]: Further to existing answers, let's clarify how transfinite recursion defines $\aleph_\alpha$ for arbitrary ordinals $\alpha$. Fr simplicity I'll identify each aleph with the least ordinal of the intended cardinality. We have $\aleph_0:=\omega$, and for any limit ordinal $\gamma\ne 0$ we have $\aleph_\gamma:=\bigcup_{\beta<\gamma}\aleph_\gamma$. Finally, $\aleph_{\alpha+1}$ is the Hartogs number of $\aleph_\alpha$, i.e. the least ordinal that cannot be injected into $\aleph_\alpha$. I recommend convincing yourself $\aleph_\alpha$ can be injected into $\aleph_\beta$ only if $\alpha=\beta\lor\alpha\in\beta$ (for ordinals we usually write this as $\alpha\le\beta$), and in particular understanding the proof that a Hartogs number exists. Hence $\aleph_\omega<\aleph_{\omega+1}<\aleph_{\omega+2}<\cdots$.