In ZFC Set Theory, "the set of all sets" famously doesn't exist since it leads to Russell's Paradox. Clearly, there are other constructions that also don't exist like "the set of all non-empty sets" for the same reason.
I want to be able to prove whether a "described" set exists or not.
For example, "the set of all sets that contain natural numbers" $(\mathcal{P}(\mathbb{N}))$ does exist, while "the set of all tuples $(a,b)$ where $a$ and $b$ are sets" does not exist (I think).
Intuitively, it seems that an injection could be used similarly to a proof about the cardinality of a set. I.e., if there is an injection from the universal set to the set $A$, then $A$ is not a set because it has "more" elements (a higher or equal cardinality) than the universal set.
For example, is this a valid proof?:
Let $U$ be the universal set
Suppose $A=\{(a,b)\ |\ a$ and $b$ are sets$\}$ exists
$\forall u\in U$, let $f(u)=(u,\emptyset)$
Then $f$ is an injection from $U$ to $A$
$\therefore A$ does not exist
Is this logic a valid way of showing that a set doesn't exist?
I don't think this proof is correct since this it uses the universal set. But is there another way to express the same logical reasoning?
Are there any "describable" but non-existent sets that cannot be shown to not exist by this logic?
And of course, is there a better general way to show that a "describable" set doesn't exist?