MATHEMATICS AND LOGIC. 157 



is precisely because it is the most important of all 

 that have been written in the new language. Besides, 

 the uninitiated can read it, thanks to an interlined 

 Italian translation. 



What gives importance to this treatise is the fact that 

 it presented the first example of those antinomies met 

 with in the study of transfinite numbers, which have 

 become, during the last few years, the despair of 

 mathematicians. The object of this note, says Signor 

 Burali-Forti, is to show that there can be two trans- 

 finite (ordinal) numbers, a and b, such that a is neither 

 equal to, greater than, nor smaller than, b. 



The reader may set his mind at rest. In order to 

 understand the considerations that will follow, he does 

 not require to know what a transfinite ordinal number is. 



Now Cantor had definitely proved that between 

 two transfinite numbers, as between two finite num- 

 bers, there can be no relation other than equality or 

 inequality in one direction or the other. But it is 

 not of the matter of this treatise that I desire to speak 

 here ; this would take me much too far from my 

 subject. 1 only wish to concern myself with the form, 

 and I ask definitely whether this form makes it gain 

 much in the way of exactness, and whether it thereby 

 compensates for the efforts it imposes upon the 

 writer and the reader. 



To begin with, we find that Signor Burali-Forti 

 defines the number i in the following manner : — 



I = t T' {Ko^(«,/i) e («c One}, 



a definition eminently fitted to give an idea of the 

 number i to people who had never heard it before. 

 I do not understand Peanian well enough to ven- 



