 Von Neumann’s system of 1925 deals with an alternative axiomatic foundation of set theory. There are two sorts of objects: objects of type II (functions, corresponding to classes) and objects of type I (arguments), linked by the application operation of a function to its arguments. The two domains partly overlap and there are objects of type I–II, corresponding to sets (as functions which can also be arguments). The fundamental axiom IV-2 then states that an object \(a\) is a proper class (i.e., it is not type I–II) if and only if the totality of its members can be mapped onto the totality of all arguments. The Burali-Forti antinomy shows that the class \(\mathbf{ON}\) of all ordinals is not a set, which implies with axiom IV-2 that there is an application of \(\mathbf{ON}\) onto the universe of all sets, and hence that the universe of sets is well-ordered. Conceptually, the system settles the problem of making Cantor’s distinction between inconsistent and consistent precise and applicable (against Hilbert’s early criticism); it also shows that global choice becomes a theorem on a suitable view of sets. Though circular objects cannot live in von Neumann’s hierarchical model of set theory, they can be found in the investigations of other mathematicians and logicians, e.g., Finsler. According to Finsler, paradoxes hinge upon circular notions, but circularity does not necessarily lead to contradictions. In particular, Finsler regards Cantor’s notion of set as intrinsically circular: sets depend on sets or on general things depending also on sets and the associated dependence graph can drive us back in a circle. For the contemporary reader, it is worth mentioning that an original intuition of Finsler 1926b is the use of graph theory for representing circular structures. So arrows are used for interpreting membership, and it is not difficult to imagine a set that has itself as unique element and more complicated circular situations (for more on Finsler’s set theory see Holmes 1996 in Other Internet Resources). Finsler 1926 applies Richard’s paradox in order to produce metamathematical results, in particular ‘formally undecidable propositions’. However, Finsler’s arguments are not conclusive and cannot be considered a proper anticipation of the Gödel incompleteness theorems (on the limit of his ideas, see the discussion in van Heijenoort, 438–440); but they show that an attentive reading of the paradox might have unexpected applications.