How can I prove that there is no set containing itself without using axiom of foundation?

Question

I've already found some similar questions in here (and other sites), but in most of the case, the use of axiom of foundation is required to complete the proof. Is there any way to prove $\not\exists x(x\in x)$, without using the axiom?

Thank you in advance.

Answer 1

You can't.

It is consistent that the axiom of foundation fails, and there are sets of the form $x=\{x\}$. More generally there are "Anti-Foundation Axioms" which assert the existence of ill-founded sets. The more famous ones are by Boffa and by Azcel.