Comparability of classes. Till now we tried to get along without the axioms Vc and Vd. We found that this is possible in number theory and analysis as well as in general set theory, even keeping in the main to the usual way of procedure.

For the considerations of the present section application of the axioms Vc, Vd is essential. Our axiomatic basis here consists of the axioms I—III, V*, Vc, and Vd. From V*, as we know, Va and Vb are derivable. We here take axiom V* in order to separate the arguments requiring the axiom of choice from the others. Instead of the two axioms V* and Vc, as was observed in Part II, V** may be taken as well.

Russell's theory of types, though probably not providing the soundest possible foundation for mathematics, follows closely the outlook of most mathematicians. The present paper is an attempt to present the theory of types in forms in which the types themselves only play a rather small part, as they do in ordinary mathematical argument. Two logical systems are described (called the “nested-type” and “concealed-type” systems). It is hoped that the ideas involved in these systems may help mathematicians to observe type theory in proofs as well as in doctrine. It will not be necessary to adopt a formal logical notation to do so.

A logical calculus Κ was defined in a previous paper and then shown in a subsequent paper to contain within itself a representation of every constructively definable subclass of expressions of a certain infinite class U of expressions, where Κ itself is one such subclass. (The former paper will be referred to as BL and the latter paper as RC.) The calculus Κ was called a “basic calculus” and its theorems were thought of as expressing the asserted propositions of a “basic logic,” that is, of a logic within which is definable every constructively definable system of logic and indeed every constructively definable class or relation. The notion of constructive definability was essentially equated with the notion of recursive enumerability.

We are inclined to believe that, by means of an argument entirely analogous to that which leads to the Richard antinomy, the notion of definability as applied to entities discussed in a formal system can easily be shown not to be itself definable in this system. It will be seen from this discussion that actually the situation is not quite so simple as it would appear at first glance. Our discussion will have a rather sketchy and informal character.