In his valuable book on mathematics and its philosophy in the

... The fact is that such a procedure is not applicable. Why? Because their definitions are not predicative and contain within such a vicious circle I already mentioned above; not predicative definitions can not be substituted to defined terms. In this condition, logistics is no longer sterile: it generates contradictions. (Jules-Henri Poincaré 1902, [10] 211, our translation.) Introduction. By common consent Russell’s antinomy is the reason for which in Zermelo–Fraenkel set theory, there is no set which comprehends all sets. Furthermore, given any set A, there is no set which contains all sets which are not members of A (in particular, there is no set which is the complement of A) ([7] 40-41). In other words, given any set A, the absolute complement of A, i.e. {x | x / ∈ A}, cannot