MATHEMATICS AND LOGIC. 159 



But I am afraid that if we asked M. Couturat what 

 two is, he would be obliged to use the word one. 



VIII. 



But let us return to the treatise of Signor Burali- 

 Forti. I said that his conclusions are in direct 

 opposition to those of Cantor. Well, one day I 

 received a visit from M. Hadamard, and the conversa- 

 tion turned upon this antinomy. 



"Does not Burali-Forti's reasoning," I said, "seem 

 to you irreproachable ? " 



" No," he answered ; " and, on the contrary, I have 

 no fault to find with Cantor's. Besides, Burali-Forti 

 had no right to speak of the whole of all the ordinal 

 numbers." 



" Excuse me, he had that right, since he could 

 always make the supposition that 



i2 = T' (No, e >). 



I should like to know who could prevent him. And 

 can we say that an object does not exist when we 

 have called it 12 ? " 



It was quite useless ; I could not convince him 

 (besides, it would have been unfortunate if I had, since 

 he was right). Was it only because I did not speak 

 Peanian with sufficient eloquence ? Possibly, but, 

 between ourselves, I do not think so. 



Thus, in spite of all this pasigraphical apparatus, 

 the question is not solved. What does this prove ? 

 So long as it is merely a question of demonstrating 

 that one is a number, pasigraphy is equal to the task ; 

 but if a difficulty presents itself, if there is an anti- 

 nomy to be resolved, pasigraphy becomes powerless. 



