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1908 B. Russell in Amer. Jrnl. Math. 30 241 (heading) The axiom of reducibility.

1910 A. N. Whitehead & B. Russell Principia Mathematica I. 168 (heading) The hierarchy of types and the axiom of reducibility.

1910 A. N. Whitehead & B. Russell Principia Mathematica I. 174 We assume, then, that every function of one variable is equivalent, for all its values, to some predicative function of the same argument. This assumption seems to be the essence of the usual assumption of classes; at any rate, it retains as much of classes as we have any use for... We will call this assumption the axiom of classes, or the axiom of reducibility.

1930 L. S. Stebbing Mod. Introd. Logic xxiii. 463 Ramsey has suggested a reconstruction of the system of Principia Mathematica in which the axiom of reducibility is no longer needed.

1942 D. D. Runes Dict. Philos. 266/2 As an indication or rough description of the axiom of reducibility, it may be said that it cancels a large part of the restrictive consequences of the prohibition against impredicative definition..and, in approximate effect, reduces the ramified theory of types to the simple theory of types.

1952 S. C. Kleene Introd. Metamath. iii. 44 To escape this outcome, Russell postulated his axiom of reducibility, which asserts that to any property belonging to an order above the lowest, there is a coextensive property (i.e. one possessed by exactly the same objects) of order 0.

1963 W. V. Quine Set Theory xi. 251 The axiom of reducibility regales us after all with attributes unspecifiable except by quantifying over attributes whose order is as high as their own.

1982 W. S. Hatcher Logical Found. Math. iv. 127 The axiom of reducibility lets us reduce the order of any term to the lowest possible order consistent with well-formedness (which is thus predicative by definition).

2006 Bull. Symbolic Logic 12 604 No matter what interpretation of the axiom of reducibility is adopted on a philosophical level, the mathematical consequences are the same.