Let V be the universe (the class of all sets), let W(0)=V, W(1) be the class of all singletons whose unique member element is a member set of W(0), and for n>0 let W(n+1) be the class of all singletons whose unique member element is a member set of W(n).
Let, for every n>=0 S(n)=W(n+1)/W(n) be the class that is the difference of the class W(n+1) and of the class W(n).
We are interested with the proposition (S): "For every set x, there exists an unique natural number n such that x is a member set of S(n)" .
Question 1: Let ZFC be our set theory; does ZFC prove (S) ?
Question 2: Suppose the answer to question 1 is YES, and let now our set theory be ZF-(I mean ZF with the omission of the axiom of Regularity (or Foundation)); does ZF- prove (S) ?
Question 3: Suppose the answer to question 2 is NO; does ZF- prove the equivalence of (S) with the axiom of regularity ?
Gérard Lang 



Finitely nested Singletons and the axiom of Regularity
I assume that you meant to write $S(n)=W(n)-W(n+1)$, rather than what you have written, since inductively one can show $W(n+1)\subseteq W(n)$. With this understanding, all the $S(n)$ are disjoint, and the question is whether every set eventually falls out, or whether there can be a set in every $W(n)$. 


In ZFC, there can be no set $x$ in every $W(n)$, since the transitive closure of such an $x$ would consist of the set containing as members the unique element of $x$, the unique element of that set, and so on, and thus have no $\in$-minimal element, contrary to the foundation axiom. So the answer to question 1 is Yes.


Meanwhile, it is relatively consistent with ZF- that there is a set $x$ which has $x=\{x\}$; for example, such sets exist under the Anti-Foundation axiom. Such a set is in every $W(n)$, and so is not in any $S(n)$, and so the answer to question 2 is No.


Question 3 is quite interesting, but I don't know the answer. You may want to add the axiom of Dependent Choices DC, a mild version of AC, for with this axiom a violation of the foundation axiom will give rise to an $\in$-descending $\omega$-sequence $x\_0,x\_1,\ldots$ with $x\_{n+1}\in x\_n$, and from such a sequence one might hope to build a set that is in every $W(n)$. But I don't see this how to complete this idea just yet...


