 It has seemed to some, beginning with Chwistek (1912) and continuing through Copi (1950) that the Axiom of Reducibility is technically faulty, leading to an inconsistency, or at least redundancy in the system of PM. Ramsey (1931) early on argued that the supposed contradiction in fact demonstrated that certain predicative functions are indefinable. Church (1976), confirms this assessment, and uses the presentation of r-types we describe here to show rigorously the limitations on what functions are definable in the system of PM.