Abstract. The conditions for proper definitions in mathematics are given, in terms of the theory of definition, on the basis of the criterions of eliminability and non-creativity. As a definition, Russell’s antinomy is a violation of the criterion of eliminability (Behmann, 1931, [1]; Bochvar, 1943, [2]). Following the path of the criterion of non-creativity, this paper develops a new analysis of Comprehension schema and, as a consequence, proof that Russell’s antinomy argumentation, despite the words of Frege himself, does not hold in Grundgesetze der Arithmetik. According to Basic Law (III), the class of classes not belonging to themselves is a class defined by a function which can not take as argument its own course of value, „ “ g ⌣ ` ´ ∀ = `ε `ε ( g(ε)) = ε → g(ε) ”« in other words, the class of classes not belonging to themselves is a class whose classes are not identical to the class itself [6].

... 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