My historical understanding (which may very well be wrong) is that initially there was naive comprehension for set construction, which required no superset. Russell's Paradox came along and blew that out of the water, and the current definition of comprehension was established for consistency.
I've recently been trying to understand the liar's paradox (this sentence is false) and I'm not at a comfortable place with it. It seems to me to be related to Russell's paradox in its self referential nature, so I've been curious about duality between sets and statements or predicates. I'm sure there is some good treatment of their relationship out there but I haven't researched that much yet (this is not my job).
What I was wondering is this... The curtailing of naive comprehension seems to me to be an unbalanced choice. Could one arrive at a similarly consistent set of axioms by restricting the class of constructible predicates and allow for comprehension without a superset?
If this question belongs in a different SE feel free to move it. Thanks!
Yes, Quine's New Foundations only allows comprehension of stratified formulas, which is some way of saying that the variables can be assigned various "levels" or "types", and a variable can only be said to be an element (or not) of something of higher types.
In particular, $x\notin x$ is not a stratified formula, so it does not define a set.
Surprisingly, $x=x$ is a stratified formula, so the set of all sets exists just fine in $\sf NF$.