Reducibility axiom

An axiom added by B. Russell and A.N. Whitehead to their ramified theory of types (cf. Types, theory of), with the aim of disposing of the stratification of concepts (see Non-predicative definition). In the ramified theory of types, sets of a given type are divided into orders. Thus, instead of the concept of a set of natural numbers, there appears that of a set of natural numbers of a given order. Here a set of natural numbers defined by formulas without using any sets belongs to the first order. If one uses in a definition a collection of sets of the first order but collections of sets of higher orders are not used, then the defined set belongs to the second order, etc. For example, if $S$ is a family of sets consisting of sets of a given order, then the set $$ M = \{ x : \exists y \in S, x \in y \} $$ must belong to the next order, since its definition contains a quantifier over sets of the given order. The reducibility axioms asserts that for each set there is a set of equal volume (that is, consisting of the same number of elements) of the first order. Thus, the reducibility axiom in fact reduces the ramified theory of types to a simple theory of types.

References

[1]	D. Hilbert, W. Ackerman, "Grundzüge der theoretischen Logik" , Dover, reprint (1946)

References

[a1]	B. Russell, A.N. Whitehead, "Principia mathematica" , 1–3 , Cambridge Univ. Press (1925–1927)