  The axioms of ZF are motivated by the second point of view – in fact by a very specific school of it: the view that the key is not mere noncircularity of \(\in\) but wellfoundedness. A set \(x\) is wellfounded if there are no infinite sequences \(\ldots x_n \in x_{n-1} \in \ldots x_3 \in x_2 \in x_1\). The axioms of ZF can usefully be thought of as arising from an attempt to axiomatise the theory of wellfounded sets.  This project has been so successful that nowadays many people believe that wellfounded sets are the only kind of set there is – if it isn’t wellfounded it isn’t a set but a hyperset perhaps. This idea is a modern rewriting of history: Set theory did not start with the idea that all sets should be wellfounded (had it done so, the axiom scheme of naïve comprehension would never have been entertained!) The idea that we should admit a candidate set as a genuine set only if promises to be wellfounded is an overreaction to the uncontroversial observation that the suspect collections that feature in the paradoxes (the universe, the Russell class) are all flagrantly illfounded, or at least would be if they existed. It is easy to jump from this insight to the conclusion that wellfoundedness of \(\in\) is the only thing that matters. The result of this jump is ZF set theory.