Does set theory allow for infinite ascending membership chains? Is every set itself a member of another set?

Question

I was thinking about a variant of Russell's paradox: Is it possible to specify a set of all objects that are themselves not members of any set?

Membership in this set would be clearly contradictory. However, there's another option: perhaps this set is empty. But if that's the case, then that would imply that every set is itself a member of another set. In other words: an infinite ascending membership chain.

Does set theory allow for this? Or is my definition -- a set of all objects that are themselves not members of any set -- disallowed by one of the axioms of modern set theory?

Answer 1

Note that for every set $x$, $$x\in\{x\},$$ where the latter set is postulated to exist per the Pairing Axiom