 Do circular objects exist in set theory? Zermelo’s view of sets as axiomatized in 1908 did not in itself exclude the possibility of self-membership. The problem was raised anew by Mirimanoff (1917a, 1917b, 1920; see also the entry  Zermelo’s axiomatization of set theory).  Once circular sets are allowed, a strengthening of extensional equality by means of a suitable isomorphism relation (bisimulation, in current terminology) is needed which essentially corresponds to the isomorphism of the trees picturing the given sets. Russell’s argument then suggests a distinction between sets of first kind, which are not isomorphic with any of their own elements, and sets of second kind which are indeed isomorphic with at least one of their elements. In the light of this distinction, Russell’s contradiction shows that the collection R of sets of first kind does not exist as a set. Indeed, a set of second kind always contains a set of second kind; hence a set of sets of first kind must be of first kind. If \(R\) were a set, it should be of first kind; but then it could not contain all sets of first kind. Mirimanoff 1917a then introduced the fundamental distinction between ordinary (well-founded) and extraordinary (non-well-founded) sets: a set \(X\) is ordinary if every \(\in\)-descending chain in \(X\) is finite; it is extraordinary, otherwise (there exist infinite \(\in\)-descending chains). It follows that all sets of the second kind are extraordinary, but the converse is not true (for instance, consider the set \(E=\{e_1, E_1\}\), where \(E_1 = \{e_1,e_2, E_2 \}\), \(E_2 =\{e_1, e_2, e_3, E_3\}\), etc.).