series 0, 1, 2, 3, fulfil the assumptions; we might prove that it satisfies assumptions 1, 2, 4, 5, but assumption 3 requires besides that 4 be an integer and that the series 0, 1, 2, 3, 4 fulfil the assumptions, and so on.
It is therefore impossible to demonstrate the assumptions for certain integers without proving them for all; we must give up proof by example.
It is necessary then to take all the consequences of our assumptions and see if they contain no contradiction.
If these consequences were finite in number, this would be easy; but they are infinite in number; they are the whole of mathematics, or at least all arithmetic.
What then is to be done? Perhaps strictly we could repeat the reasoning of number III.
But as we have said, this reasoning is complete induction, and it is precisely the principle of complete induction whose justification would be the point in question.
VI
The Logic of Hilbert
I come now to the capital work of Hilbert which he communicated to the Congress of Mathematicians at Heidelberg, and of which a French translation by M. Pierre Boutroux appeared in l'Enseignement mathématique, while an English translation due to Halsted appeared in The Monist.1313 In this work, which contains profound thoughts, the author's aim is analogous to that of Russell, but on many points he diverges from his predecessor.
"But," he says (Monist, p. 340), "on attentive consideration we become aware that in the usual exposition of the laws of logic certain fundamental concepts of arithmetic are already employed; for example, the concept of the aggregate, in part also the concept of number.
"We fall thus into a vicious circle and therefore to avoid paradoxes a partly simultaneous