THE POSITIVE THEORY OF INFINITY 205 



the second place, it may be admitted that the definition, 

 like all definitions, is to a certain extent arbitrary. In 

 the case of the small finite numbers, such as 2 and 3, it 

 would be possible to frame definitions more nearly in 

 accordance with our unanalysed feeling of what we mean ; 

 but the method of such definitions would lack uniformity, 

 and would be found to fail sooner or later at latest when 

 we reached infinite numbers. 



In the third place, the real desideratum about such a 

 definition as that of number is not that it should repre- 

 sent as nearly as possible the ideas of those who have not 

 gone through the analysis required in order to reach a 

 definition, but that it should give us objects having the 

 requisite properties. Numbers, in fact, must satisfy the 

 formulae of arithmetic ; any indubitable set of objects 

 fulfilling this requirement may be called numbers. So 

 far, the simplest set known to fulfil this requirement is 

 the set introduced by the above definition. In comparison 

 with this merit, the question whether the objects to which 

 the definition applies are like or unlike the vague ideas 

 of numbers entertained by those who cannot give a 

 definition, is one of very little importance. All the 

 important requirements are fulfilled by the above defini- 

 tion, and the sense of oddity which is at first unavoidable 

 will be found to wear off very quickly with the growth 

 of familiarity. 



There is, however, a certain logical doctrine which may 

 be thought to form an objection to the above definition 

 of numbers as classes of classes I mean the doctrine 

 that there are no such objects as classes at all. It might 

 be thought that this doctrine would make havoc of a 

 theory which reduces numbers to classes, and of the 

 many other theories in which we have made use of classes. 

 This, however, would be a mistake : none of these theories 



