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



Finite T-uples and the axiom of Regularity
Question 3 also has a negative answer (Joel Hamkins has already answered the first two questions).


The model $V(a,b,c)$ described in detail in my answer to an analogous [question](https://mathoverflow.net/questions/67310/finitely-nested-singletons-and-the-axiom-of-regularity) works here as well since *$V(a,b,c)$ satisfies $ZF+T$, but **not** Foundation*.


The reason that $T$ holds in $V(a,b,c)$ is similar to the reason that $S$ holds in the other question: any infinite descending epsilon sequence must eventually hit one of the elements $a$, $b$, or $c$, none of which is a Kuratowski-ordered pair (indeed, they are not ordered pairs in the sense of Wiener either).


