In the appendix B to the second edition of PM, which was written by Russell, there is a technical discussion of the consequences of abandoning the axiom of reducibility. A faulty proof is proposed to show that the principle of Induction can be derived without using the axiom of reducibilty in a modified theory of types (see Linsky 2011). As Russell points out, however, it is not possible to define real numbers using “Dedekindian” classes of rational numbers without assuming the axiom of reducibi