LAST EFFORTS OF LOGISTICIANS. 185 



antinomies to which I have already made frequent 

 allusion. Cantor thought it possible to construct a 

 Science of the Infinite. Others have advanced further 

 along the path he had opened, but they very soon ran 

 against strange contradictions. These antinomies are 

 already numerous, but the most celebrated are : — 



1. Burali-Forti's antinomy. 



2. The Zermelo-Konig antinomy. 



3. Richard's antinomy. 



Cantor had demonstrated that ordinal numbers (it 

 is a question of transfinite ordinal numbers, a new 

 notion introduced by him) can be arranged in a lineal 

 series ; that is to say, that of two unequal ordinal 

 numbers, there is always one that is smaller than the 

 other. Burali-Forti demonstrates the contrary ; and 

 indeed, as he says in substance, if we could arrange all 

 the ordinal numbers in a lineal series, this series 

 would define an ordinal number that would be 

 greater than all the others, to which we could then 

 add I and so obtain yet another ordinal number 

 which would be still greater. And this is contra- 

 dictory. 



We will return later to the Zermelo-Konig anti- 

 nomy, which is of a somewhat different nature. 

 Richard's antinomy is as follows {Reviie ghierale des 

 Sciences, June 30, 1905). Let us consider all the 

 decimal numbers that can be defined with the help of 

 a finite number of words. These decimal numbers form 

 an aggregate E, and it is easy to sec that this aggregate 

 is denumerable — that is to say, that it is possible to 

 number "C^^ decimal numbers of this aggregate from one 

 to infinity. Suppose the numeration effected, and let 



