LAST EFFORTS OF LOGISTICIANS. 191 



Upon what condition will this latter definition, 

 which plays an essential part in Whitehead's demon- 

 stration, be " predicative" and consequently acceptable ? 



Following upon what has been said above, we must 

 understand by all recurrent classes all those whose 

 definition does not contain the notion of inductive 

 number ; otherwise we shall be involved in the vicious 

 circle which engendered the antinomies. 



Now, Whitehead has not taken this precaution. 



Whitehead's argument is therefore vicious ; it is the 

 same that led to the antinomies. It was illegitimate 

 when it gave untrue results, and it remains illegitimate 

 when it leads by chance to a true result. 



A definition which contains a vicious circle defines 

 nothing. It is of no use to say we are sure, whatever 

 be the meaning given to our definition, that there is 

 at least zero which belongs to the class of inductive 

 numbers. It is not a question of knowing whether 

 this class is empty, but whether it can be rigidly 

 delimited. A " non-predicative class" is not an empty 

 class, but a class with uncertain boundaries. 



It is unnecessary to add that this particular objection 

 does not invalidate the general objections that apply 

 to all the demonstrations. 



IX. 



Signor Burali-Forti has given another demonstration 

 in his article " Le Classi finite" {Atti di Torino, 

 Vol. xxxii). But he is obliged to admit two postulates : 



The first is that there exists always at least one 

 infinite class. 



The second is stated thus : — 



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