 Russell ramified his theory of types in order to avoid explicitly impredicative definitions (against which definitions Poincaré influentially inveighed). Russell then found himself hamstrung, unable to derive certain desired mathematical results. Among these results were Cantor’s Theorem, and the theorem of real analysis which states that every set X of real numbers that is bounded above has a least upper bound of the same order as the real numbers in X. Ramified type theory appeared powerless to prove these results. So Russell, in pragmatist spirit, introduced the Axiom of Reducibility simply in order to get things done. 