Do all paradoxes of naive set theory have something in common?

If P(x) is the formula "x ∉ x", then …

the assumption that a set h has P(x) purity …

(i.e. the assumption that for all t, if t∈h then P(t))

… implies that there exists a set k, where k exhibits the fact that h isn’t comprehensive with respect to P(x). In other words, we could say that k is an exemplar of the deficiency of h with respect to P(x).

Formally, we could define "h isn’t comprehensive with respect to P(x)" to mean that, for some t, P(t) is true and t∉h.

Observe, in particular, that if P(x) is the formula "x ∉ x", then we can choose k to simply be h.

Now, if the above is merely a tautology, then the point of interest is the possibility of having a way to formally represent the functional relationship that maps h ↦ k.

Question: What apparatus could be used to formalize the mapping h ↦ k?

First follow-up question: Could the following be a first draft for an intuitively plausible schema of set existence: if there doesn’t exist m such that, for all x, (x∈m iff P(x)), then there is a mapping h ↦ k, such that if h has P(x) purity, then k is an exemplar of the fact that h isn’t comprehensive with respect to P(x)?

Second follow-up question: considering the contrapositive of the above schema of set existence, could we use reasoning analogous to what is used in the paradoxes of naive set theory to demonstrate that some particular mappings h ↦ k don’t exist, and conclude that there exists m such that, for all x, (x∈m iff P(x))?

Third follow-up question: Are there techniques for generating an unlimited number and variety of paradoxes of naive set theory that could be used to generate examples ad lib of non-existent mappings h ↦ k?

Comment: perhaps any attempt to create apparatus within set theory to give formal existence to a mapping h ↦ k will inevitably produce contradictions, if for every value h that has P(x) purity, the value k is an exemplar of the fact that h isn’t comprehensive with respect to P(x), and there doesn’t exist m such that, for all x, (x∈m iff P(x)).

It is conceivable that that is the correct answer to the question. If that is your answer, then it would be helpful if you could provide some justification for it.