Reducibility, axiom of
Reducibility, axiom of
An axiom which (or some substitute) is necessary in connection with the ramified theory of types (q.v.) if that theory is to be adequate for classical mathematics, but the admissibility of which has been much disputed (see Paradoxes, logical). An exact statement of the axiom can be made only in the context of a detailed formulation of the ramified theory of types — which will not here be undertaken. As an indication or rough description of the axiom of reducibility, it may be said that it cancels a large part of ihe restrictive consequences of the prohibition against impredicative definition (q.v.) and, in approximate effect, reduces the ramified theory of types to the simple theory of types (for the latter see Logic, formal, 6). — A.C.