On: Set Theory and Foundations of Mathematics

[defen]

> universal set

But now you're talking about non-standard set theory[1] which is fine but you are kind of side-stepping the issue.

[1] https://en.wikipedia.org/wiki/Axiom_of_regularity

[Shaniqua → defen]

Well, it's not that difficult to fix it in ZF.

If you define S to be S = {x in U: x not in x}, then it simply means S is not in U.

[defen → Shaniqua]

There is no U in ZF so I really don't understand what you are saying.

[Shaniqua → defen]

You can call U anything you want. It's by axiom schema of specification: for any set A, there some set B with a set C in B iff C in A.