LAST EFFORTS OF LOGISTICIANS. 195 



change. If the number of the objects is indefinite, 

 that is to say if we are constantly liable to find new 

 and unforeseen objects springing up, it may happen 

 that the appearance of a new object will oblige us to 

 modify the classification, and it is thus that we are 

 exposed to the antinomies. 



There is no actual infinity. The Cantorians forgot 

 this, and so fell into contradiction. It is true that 

 Cantorism has been useful, but that was when it was 

 applied to a real problem, whose terms were clearly 

 defined, and then it was possible to advance without 

 danger. 



Like the Cantorians, the logisticians have forgotten 

 the fact, and they have met with the same difficulties. 

 But it is a question whether they took this path by 

 accident or whether it was a necessity for them. 



In my view, there is no doubt about the matter ; 

 belief in an actual infinity is essential in the Russellian 

 logistic, and this is exactly what distinguishes it from 

 the Hilbertian logistic. Hilbert takes the point of 

 view of extension precisely in order to avoid the 

 Cantorian antinomies. Russell takes the point of 

 view of comprehension, and consequently for him the 

 genus is prior to the species, and the suvmimn genus 

 prior to all. This would involve no difficulty if the 

 summuni genus were finite ; but if it is infinite, it is 

 necessary to place the infinite before the finite — that is 

 to say, to regard the infinite as actual. 



And we have not only infinite classes ; when we 

 pass from the genus to the species by restricting the 

 concept by new conditions, the number of these 

 conditions is still infinite, for they generally express 

 that the object under consideration is in such and 



