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:
1, 2, 3, 4, 5, 6, 7, 8, 1, 1
As we see, N is not equal to the nth 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