assures that there is a predicative property bH true of the same rational numbers as b. Since the reals are predicative, hence of the same order as bH, it turns out that bH is a real number, and hence that S has a least upper bound after all, as required by the classical theorem. The general role of reducibility is thus to undo the draconian mathematical effects of ramification without undermining its capacity to fend off the semantic paradoxes. See also HIERARCHY , PARADOX, RUSSEL. C.M.