 Poincaré (1906b) took the contradictions as grounds to defend an intuitionistic, Kantian point of view. According to him, number-theoretic induction and the axiom of choice constitute independent intuitions, truly synthetic a priori judgements. He then argues against Russell’s logical definition of natural numbers as those numbers which belong to all recurrent classes (those classes which contain 0 and are closed under successor). His objection is that the definition is not admissible since it refers essentially to a totality to which the class to be defined belongs—the definition is impredicative—and hence it is to be regarded as circular. But, according to Poincaré, mathematical objects do not exist without a proper definition, and a proper definition must be predicative, i.e., it must avoid vicious circles; Poincaré thus somewhat extended Richard’s diagnosis.