 Mirimanoff’s work is also important for the foundations of set theory. He introduced the notion of ordinal rank for ordinary sets and he noticed that ordinary sets can be arranged in a cumulative hierarchy, indexed by their ranks. However, the existence of a cumulative structure of ordinary sets is not considered as a ground for excluding extraordinary sets. Mirimanoff (pp. 212–213) explicitly points to the use of extraordinary sets for modelling mirror-like situations. He mentions the case of a book \(B\) whose covering is decorated with a picture \(J\) representing two children glancing at the same book, i.e. to the picture \(J1\) of \(B\). In \(J_1\) one could perceive again the two children and the picture \(J_{11}\) of the book in perspective. Now \(J\) can be regarded as a set including as elements the two children \(e_1\) and \(e_2\), and the picture \(J_1\) of \(J\), which in turn decomposes into the pictures \(e_{11}, e_{22}\) of \(e_1\) and \(e_2\), and the picture \(J_{11}\) of \(J_1\), and so on ad infinitum, Now \(J\) is isomorphic to one of its elements, i.e. \(J_1: J\) can be considered as a set of second kind and hence extraordinary. This non-mathematical example is suggestive of later developments, i.e., the theory of non-well founded sets and its recent applications to semantics. A set \(E\) of second kind is also assimilated to an impredicative collection in the sense of Poincaré (Mirimanoff 1920, p. 34) because of its circularity: indeed, \(E\) is given by a condition \(E=(y, z,\ldots ,a, b, c,\ldots)\), where \(y, z,\ldots\) depend on \(E\).