Is it possible for any set to be an element of itself?

Question

Is it possible for any set to be an element of itself? I want brief explanation for whether it's possible or not.

Answer 1

In ZF, the answer is no. The reason is Axiom of regularity is one of the axioms there.

Answer 2

It depends on the axioms.

In $\mathsf {ZF}$, as said before, this is not possible.

According to other axiomatizations, like Quine's New Foundations, we can have the universal set $V$, and thus:

$V \in V$.