190 SCIENCE AND METHOD. 



the definition of E would contain a vicious circle, for 

 we cannot define E by the aggregate E itself. 



Now we have defined N by a finite number of 

 words, it is true, but only with the help of the notion 

 of the aggregate E, and that is the reason why N does 

 not form a part of E. 



In the example chosen by M. Richard, the con- 

 clusion is presented with complete evidence, and the 

 evidence becomes the more apparent on a reference to 

 the actual text of the letter. But the same explana- 

 tion serves for the other antinomies, as may be easily 

 verified. 



Thus the definitions that must be regarded as non- 

 predicative are those which contain a vicious circle. 

 The above examples show sufficiently clearly what 

 I mean by this. Is this what Mr. Russell calls 

 " zigzagginess " ? I merely ask the question without 

 answering it. 



VIII. 



The Demonstrations of the Principle 

 OF Induction. 



We will now examine the so-called demonstrations 

 of the principle of induction, and more particularly 

 those of Mr. Whitehead and Signor Burali-Forti. 



And first we will speak of Whitehead's, availing our- 

 selves of some new denominations happily introduced 

 by Mr. Russell in his recent treatise. 



We will call recurrent class every class of numbers 

 that includes zero, and also includes « + 1 if it 

 includes n. 



We will call inductive number every number which 

 forms a part of all recurrent classes. 



