The Non-Finitizability of Impredicative Principles

Abstract

An impredicative class is a class definable only by reference to a totality to which the class itself belongs. Usual systems of set theory contain principles which provide us with infinitely many ways of generating such classes. We shall prove that we cannot in general replace these by finitely many, as we can in the case of predicative classes. In order to render the proof and what we prove clearer, we shall concentrate our attention on a rather special simple case. It will be obvious that other cases can be treated similarly. Let L be the system determined by Bernays's axioms' I-III and Va. This is a system with finitely many axioms. Roughly speaking, in L we are assured that all finite classes obtained from the empty class by applying (any finite number of times) the operations of forming unit classes and sum classes are sets (membership-eligible classes), that every subclass of a set is a set, and that we have all predicative classes of these sets. Bernays has shown in detail that, by identifying natural numbers with certain sets of L, we can obtain the usual number theory in L. For our purpose, we may describe L in the following manner. L contains the first-order predicate calculus (quantification theory) for one kind of variables X, Y, etc. (whose range consists of all classes) and an additional two-place predicate e (membership). Since sets are merely classes which can be members of classes, variables ranging over sets can be introduced by contextual definitions such as: (x)+ stands for (X) ((2 Y)(X E Y) D OX). It follows immediately that if (X)+X then (x)+x. Let us agree further that X = Y stands for (x) (x e X = x e Y) The axioms of L can be stated thus :2