  Because of these problems, Russell and Whitehead introduced in the first edition of Principia Mathematica the following reducibility axiom: the hierarchy of predicates, first-order, second-order, etc., collapses at level 1. This means that for any predicate of any order, there is a predicate of the first-order level which is equivalent to it. In the case of equality for instance, we postulate a first-order relation “\(a=b\)” which is equivalent to “\(a\) satisfies all properties that \(b\) satisfies”. The motivation for this axiom was purely pragmatic. Without it, all basic mathematical notions, like real or natural numbers are stratified into different orders. Also, despite the apparent circularity of impredicative definitions, the axiom of reducibility does not seem to lead to inconsistencies. 