The Nature of the Axiom of Reducibility
The so-called axiom of reducibility plays a prominent part in the reconstruction of mathematics, as carried out in Principia Mathematical,1 the basic work of the logical school. The axiom says: for any propositional function \(\varphi \hat x\) there is always a predicative propositional function \(\psi !\hat x\) which is formally equivalent to \(\varphi \hat x\).2 The role of the axiom of reducibility is to make possible the transition from any function to a predicative function and thereby to remove the difficulties entailed by the setting-up of the so-called ramified hierarchy.3 Within the framework of Principia Mathematica, such an axiom is indispensable. Without it or some equivalent axiom, the theory of real numbers, as developed there, would collapse, and with it the foundations of analysis. On the other hand, in setting up the axiom. Russell and Whitehead had felt instinctively that it had a different character from the other axioms of logic and that the same degree of certainly could not be ascribed to it; but since the real nature of the axiom remained obscure, the authors were content to arrange their work in such a way that it was possible to recognize at each point which propositions depended on the axiom of reducibility and which not.
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