i86 SCIENCE AND METHOD. 



us define a number N in the following manner. If 

 the n"' decimal of the «"' number of the aggregate E is 



0, I, 2, 3, 4, 5, 6, 7, 8, or 9, 

 the «'* decimal of N will be 



1, 2, 3, 4, 5, 6, 7, 8, I, or I. 



As we see, N is not equal to the n'" number of E, 

 and since n is any chance number, N does not belong 

 to E, and yet N should belong to this aggregate, since 

 we have defined it in a finite number of words. 



We shall see further on that M. Richard himself 

 has, with much acuteness, given the explanation of his 

 paradox, and that his explanation can be extended, 

 mutatis vmtandis, to the other paradoxes of like 

 nature. Mr. Russell quotes another rather amusing 

 antinomy : 



What is the smallest whole number that cannot be 

 defined in a sentence formed of less than a hundred 

 English words ? 



This number exists, and, indeed, the number of 

 numbers capable of being defined by such a sentence 

 is evidently finite, since the number of words in the 

 English language is not infinite. Therefore among 

 them there will be one that is smaller than all the 

 others. 



On the other hand the number does not exist, for 

 its definition involves contradiction. The number, in 

 fact, is found to be defined by the sentence in italics, 

 which is formed of less than a hundred English words, 

 and, by definition, the number must not be capable 

 of being defined by such a sentence. 



