Questions about Aleph-Aleph-Null 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.
 A: Ordinals are not cardinals.
Recall Hilbert's hotel. Where you have infinitely many rooms, one for each natural number, and they are all full. And there's a party, with all the guests invited. At some point, after so many drinks, people need to use the restroom.
So someone goes in, and immediately after another person comes and stands in line. They only have to wait for the person inside to come out, so they have $0$ people in front of them, and then another person comes and they only have to wait for $1$ person in front of them, and then another and another and so on. That's fine. But the person in the bathroom had passed out, unfortunately, and everyone is so polite, so they just wait quietly. And the queue gets longer.
Let's for concreteness sake, point out that only people who stay in rooms with an even room number go to the toilet. The others are just fine holding it in. Now for every given $n$, there is someone in the queue which needs to wait for at least $n$ people. The queue is infinite. But it's fine, since each person has only to wait a finite amount of time for their turn.
But what's this now? The person in room $3$ has to use the toilet as well. But they cannot cut in the line, that would be impolite. So they stand at the back. Well. There were $\aleph_0$ people in the queue, that's how many, and we added just one more, so there are still $\aleph_0$ people waiting in line. But now we have one person who has to wait for infinitely many people to go before them. So the queue is ordered in a brand new way. If they were lucky and someone decided to let them cut in line, then the queue would have looked the same, just from some point on people would have to wait just one more person to go first.
This is not what happened, though. So the queue looks different. Well, now we continue, all the people in room numbers which are powers of $3$ start to follow. And at some point we get to a queue which looks like two copies of the natural numbers stitched up. And then the bloke from room $5$ joins the line, and he has to two for two infinite queues before their turn. And so on and so forth.

Okay, what's the point of all that?
The point is that for finite queues the question of "how many people" and "how is the queue ordered" are the same question. So adding one person does not matter where this person was added to the queue. But when the queue was infinite, adding one person at the end or adding it to the middle would very much change the queue's order. So "how many" is no longer the same as "how long is the queue".
When iterating an operation transfinitely many times, e.g. by taking power sets or cardinal successors, we work successively. This creates a queue-like structure of cardinals. The first, the second, etc., which are ordinal numbers, they talk about order.
So once you go through the finite ones, you have to move to infinite ordinals, not to infinite cardinals. As such $\omega$ is the appropriate notation, since it denotes an ordinal, rather than $\aleph_0$ which denotes a cardinal.
Between $\aleph_\omega$ and $\aleph_{\omega+1}$ there are similarities: both have infinitely many [infinite] cardinals smaller than themselves. But it is not the same, exactly because we are dealing with the question "how are these ordered" rather than "how many are there".
Note that $\aleph$ numbers are not defined by power sets, these are $\beth$ numbers (Beth is the second letter of the Hebrew alphabet, whereas Aleph is the first one). But this is irrelevant to your actual question.
A: 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 $\aleph_\omega<\aleph_{\omega+1}<\aleph_{\omega+2}<\cdots$.
A: The reason it should really be $\aleph_\omega$ instead of $\aleph_{\aleph_0}$ is that we think of $\aleph_\alpha$ for ordinals $\alpha$. Yes, $\aleph_0=\omega$, but we write $\omega$ when we think of it as an ordinal instead of a cardinal.
The rest of  your question is based on a fundamental misunderstanding: $\aleph_{\alpha+1}$ is not the cardinality of the power set of $\aleph_\alpha$; what it is is the smallest cardinal larger than $\aleph_\alpha$.
