Thread: Set of all cardinal numbers

1. I've heard that there is no set that contains all the cardinal numbers {0, 1, 2, ... Aleph_0, Aleph_1, Aleph_2, ...}
For this to be true, obviously there must be infinite cardinals other than those of Aleph_n with n being a finite cardinal. Otherwise I just described the set here.
Is there a proof of this? I've looked it up and the best I've ever seen was that the cardinality of such a set would be bigger than any element in the set, but I fail to see why this has to be the case.

2.

3. There are more cardinal numbers than just the aleph numbers (List of large cardinal properties - Wikipedia, the free encyclopedia), at least in most set theories.

Also, a group of objects satisfying a given property is not a set by default. Allowing such constructions leads to things like Russel's paradox which is why people started trying to develop the axioms of set theory in the first place. (That said, I think that all the ordinals up through the aleph numbers might still be a set. Maybe even all the ordinals up through any given stopping point, but I don't think I'd bet on that.)

4. If S represented the set of all sets, and C represented the set of all cardinal numbers, I could say that cardinality is a function from S to C. But neither S nor C exists. So that might make it hard to formalize the idea of cardinality. I need to know this because I want to formalize mathematics myself. I understand by Godel's incompleteness theorems that it will never be finished. But that doesn't bother me.
And I looked at that Wikipedia page. It is the most interesting thing I have seen in a long time. This only makes me ask even more questions. In particular, for any set of cardinal numbers, is there guaranteed a cardinal number larger than any element in that set?
EDIT: Your hypothesis is true, I can prove it.
Let x be any cardinal number. So there must be a set X with cardinality x. Then take the power set of X. This set contains an element with every cardinality up to x, making it clear there is a set containing all cardinal numbers up to our given number x.

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