Hi

The axiom of regularity just states that a set has to contain an element which has no common element with the set.

No, sets of this form exist: if does not contain (as elements) nor then and there is no problem.Does it mean we cannot have set such that: A = {B, C} and B = {C, D}

A consequence of this axiom is that for any set that is impossible for any sets and more generally, there can't be any infinite downward -sequence between elements of a set.

But this axiom does not belong to the theory Z, and I've heard that some researchers are exploring set theories with its negation.