_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:
1. The Burali-Forti antinomy;
2. The Zermelo-König antinomy;
3. 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 antinomy[15] 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 _n_th decimal of the _n_th number of the assemblage _E_ is
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
the _n_th decimal of _N_ shall be:
1, 2, 3, 4, 5, 6, 7, 8, 1, 1
[15] _Revue générale des sciences_, June 30, 1905.
As we see, _N_ is not equal to the _n_th number of _E_, and as _n_ is arbitrary, _N_ does not appertain to _E_ and yet _N_ should belong to this assemblage since we have defined it with a finite number of words.
We shall later see that M. Richard has himself given with much sagacity the explanation of his paradox and that this extends, _mutatis mutandis_, to the other like paradoxes. Again, Russell cites another quite amusing paradox: _What is the least whole number which can not be defined by a phrase composed of less than a hundred English words_?
This number exists; and in fact the numbers capable of being defined by a like phrase are evidently finite in number since the words of the English language are not infinite in number. Therefore among them will be one less than all the others. And, on the other hand, this number does not exist, because its definition implies contradiction. This number, in fact, is defined by the phrase in italics which is composed of less than a hundred English words; and by definition this number should not be capable of definition by a like phrase.
