That's a very good response.
Logically I would state that the reason why aleph-null makes no sense is because like both aleph-null and the numbers on the list are infinity, then to say that aleph-null is in any way different from the numbers on the list is to state that the numbers on the list have an ending. This is because for them to actually be different is for both numbers to end in a difference. Such statement may sound illogical at first glance because no matter what number appears it will always be different, however the point is that if we think about it aleph-null is still impossible to fully become different.
All the notation that comes after aleph-null is to assume the difference. Here is a mathematical intepretation of my reasoning.
j = {0, 1, 2, 3, 4 ... M }
N = {1, 2, 3, 4, 5 ... X }
j = numbers on the list
M = hyperreal of j
N = Aleph-null
X= Statement of difference
however
N = {1, 2, 3, 4, 5 ... X },
N = {1, 2, 3, 4, 5 ... M ... X2... },
N = {1, 2, 3, 4, 5 ... M ... M2 ... X3 ...},
To say that aleph is different from j is to say that j ends at some M for it to become X. It will never do such a thing.
Now, the real numbers are greater than the naturals because they're both defined by themselves to be so. In this case aleph is defined to exist as long as it satisfies the difference it has with j. It doesn't have one because it cannot happen due to the nature of what is infinity.
Mathematically you can reduce all this into cardinality but it throws out the window the sense of infinity in any physical and reasonable sense, and so it defines things in a finite way similar to a hyperreal. Stating that it exist if one day M stops growing to become X.
Even then the argument behind a hyperreal is more logical then that of aleph-null. Because the hyperreal is taken into account to never be n. And so, to be able to stay at some n.
The difference X involves taking some n of j to be x. Which is to state that M is an n. No logos my buddies, what the heck is going on?
With this I say that these mathematics are physically inconsistent. So you'll never see anything like this in the physical world. To think this is amazing is to be amazed at 2 + 2 = 3 but in a more complex and sophisticated way