It is toward the no-classes theory that Russell finally inclines. Be that as it may, logistic is to be remade and it is not clear how much of it can be saved. Needless to add that Cantorism and logistic are alone under consideration; real mathematics, that which is good for something, may continue to develop in accordance with its own principles without bothering about the storms which rage outside it, and go on step by step with its usual conquests which are final and which it never has to abandon.
VII
The True Solution
What choice ought we to make among these different theories? It seems to me that the solution is contained in a letter of M. Richard of which I have spoken above, to be found in the Revue générale des sciences of June 30, 1905. After having set forth the antinomy we have called Richard's antinomy, he gives its explanation. Recall what has already been said of this antinomy. E is the aggregate of all the numbers definable by a finite number of words, without introducing the notion of the aggregate E itself. Else the definition of E would contain a vicious circle; we must not define E by the aggregate E itself.
Now we have defined N with a finite number of words, it is true, but with the aid of the notion of the aggregate E. And this is why N is not part of E. In the example selected by M. Richard, the conclusion presents itself with complete evidence and the evidence will appear still stronger on consulting the text of the letter itself. But the same explanation holds good for the other antinomies, as is easily verified. Thus the definitions which should be regarded as not predicative are those which contain a vicious circle. And the preceding examples sufficiently show what I mean by that. Is it this which Russell calls the 'zigzaginess'? I put the question without answering it.
VIII
The Demonstrations of the Principle of Induction
Let us now examine the pretended demonstrations of the principle of induction and in particular those of Whitehead and of Burali-Forti.
We shall speak of Whitehead's first, and take advantage of certain new terms happily introduced by Russell in his recent memoir. Call recurrent class every class containing zero, and containing n + 1 if it contains n. Call inductive number every number which is a part of all the recurrent classes. Upon what condition will this latter definition, which plays an essential rôle in Whitehead's proof, be 'predicative' and consequently acceptable?
In accordance with what has been said, it is necessary to understand by all the recurrent classes, all those in whose definition the notion of inductive number does not enter. Else we fall again upon the vicious circle which has engendered the antinomies.