 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 reducibility. (The thesis that every class of reals with an upper bound has a real number as its least upper bound, discussed above, would not be provable.) As a result, Russell says “analysis would collapse”. In all of this discussion, however, Russell does not indicate what would replace the definition of Identity in ∗13, which so crucially depends on the axiom of reducibility.