View source for Reducibility axiom

An axiom added by B. Russell and A.N. Whitehead to their ramified theory of types (cf. [[Types, theory of|Types, theory of]]), with the aim of disposing of the stratification of concepts (see [[Non-predicative definition|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080390/r0803901.png" /> is a family of sets consisting of sets of a given order, then the set

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080390/r0803902.png" /></td> </tr></table>

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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Hilbert,   W. Ackerman,   "Grundzüge der theoretischen Logik" , Dover, reprint  (1946)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Russell,   A.N. Whitehead,   "Principia mathematica" , '''1–3''' , Cambridge Univ. Press  (1925–1927)</TD></TR></table>