192 SCIENCE AND METHOD. 



The first postulate is no more evident than the 

 principle to be demonstrated. The second is not 

 only not evident, but it is untrue, as Mr. Whitehead 

 has shown, as, moreover, the veriest schoolboy could 

 have seen at the first glance if the axiom had been 

 stated in intelligible language, since it means : the 

 number of combinations that can be formed with 

 several objects is smaller than the number of those 

 objects. 



X. 



Zermelo's Axiom. 



In a celebrated demonstration, Signor Zermelo 

 relies on the following axiom : 



In an aggregate of any kind (or even in each of 

 the aggregates of an aggregate of aggregates) we 

 can always select one element at random (even if 

 the aggregate of aggregates contains an infinity 

 of aggregates). 



This axiom had been applied a thousand times with- 

 out being stated, but as soon as it was stated, it raised 

 doubts. Some mathematicians, like M. Borel, rejected 

 it resolutely, while others admitted it. Let us see what 

 Mr. Russell thinks of it according to his last article. 



He pronounces no opinion, but the considerations 

 which he gives are most suggestive. 



To begin with a picturesque example, suppose that 

 we have as many pairs of boots as there are whole 

 numbers, so that we can number the pairs from i to 

 infinity, how many boots shall we have? Will the 

 number of boots be equal to the number of pairs.? 

 It will be so if, in each pair, the right boot is dis- 



