 The paradoxes of the theory of sets are resolved by reducing assertions about sets to assertions about propositional functions. The restriction that a function of one type cannot apply to a function of the same type is enough to block the paradoxes. Thus the distinction between individuals, functions of individuals, and functions of such functions, categorized by what came to be called “simple theory of types” is enough for the purposes of reducing mathematics to classes, and so to logic. The idea that the full theory of types was not needed to resolve the mathematical or set theoretical paradoxes was proposed by Chwistek (1921) and Ramsey (1931), and led to the later introduction of the terms “ramified theory of types” and “simple theory of types” that will be used in this entry.