Does it mean that we must be sure of not meeting contradiction after a finite number of propositions, the finite number being by definition that which has all properties of recurrent nature, so that if one of these properties fails—if, for instance, we come upon a contradiction—we shall agree to say that the number in question is not finite? In other words, do we mean that we must be sure not to meet contradictions, on condition of agreeing to stop just when we are about to encounter one? To state such a proposition is enough to condemn it.
So, Hilbert's reasoning not only assumes the principle of induction, but it supposes that this principle is given us not as a simple definition, but as a synthetic judgment a priori.
To sum up:
A demonstration is necessary.
The only demonstration possible is the proof by recurrence.
This is legitimate only if we admit the principle of induction and if we regard it not as a definition but as a synthetic judgment.
V
The Cantor Antinomies
Now to examine Russell's new memoir. This memoir was written with the view to conquer the difficulties raised by those Cantor antinomies to which frequent allusion has already been made. Cantor thought he could construct a science of the infinite; others went on in the way he opened, but they soon ran foul of strange contradictions. These antinomies are already numerous, but the most celebrated are:
- The Burali-Forti antinomy;
- The Zermelo-König antinomy;
- The Richard antinomy.
Cantor proved that the ordinal numbers (the question is of transfinite ordinal numbers, a new notion introduced by him) can be ranged in a linear series; that is to say that of two unequal ordinals one is always less than the other. Burali-Forti proves the contrary; and in fact he says in substance that if one could range all the ordinals in a linear series, this series would define an ordinal greater than all the others; we could afterwards adjoin 1 and would obtain again an ordinal which would be still greater, and this is contradictory.
We shall return later to the Zermelo-König antinomy which is of a slightly different nature. The Richard antinomy1515 is as follows: Consider all the decimal numbers definable by a finite number of words; these decimal numbers form an aggregate E, and it is easy to see that this aggregate is countable, that is to say we can number the different decimal numbers of this assemblage from 1 to infinity. Suppose the numbering effected, and define a number N as follows: If the nth decimal of the nth number of the assemblage E is
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
the nth decimal of N shall be: