Transjinite Numbers. 457 



The argument must therefore be condemned as unsound. 

 The missing link in the proof can be supplied as follows : — 

 Let M be an aggregate of cardinal number 8 and let P 

 be an aggregate of cardinal number u having no term in 

 common with M, and let N be the logical sum of M and P. 

 Then the cardinal number of N is ft+t). Since 3+6— 3 

 we can define a one to one correlation between the terms of 

 M and the terms of N. Let this be done, and let P x denote 

 the aggregate of terms of M which are correlated with the 

 terms of P in N ; and let P 2 denote the aggregate of terms of 

 M which are correlated with the terms of P 2 in N, and so on ; 

 so that P v+ i denotes the aggregate of terms of M, which are 

 correlated with the terms of P v in N. Each of the aggregates 

 P„ is by definition entirely comprised in M, and each is of 

 cardinal number b *• 



It only remains to prove that no two of the aggregates P„ 

 have any term in common. If it be possible, let the term x 

 be common to P„ and P^ (v >//,), and let y be the term of N 

 with which x as a term of M is correlated. Then, since the 

 correlation is one-one, y must be common to P v _i and P^-i, 

 and if z be the term of N with which y as a term of M is 

 correlated, then z must be common to P„_2 and P^-2 ; by 

 proceeding in this way we prove ultimately that Pj,-^ and P 

 have a term in common, but this cannot be because P has no 

 term in common with M. This completes the proof. 



Mr. Jourdain's proof that $^-=$ l appears to me to contain 

 an unsound inference of a somewhat similar character. He 

 proves that the terms of a double series ( a W/s)> sucn that 

 a-\-/3<x {x — r number of the second class) can be correlated 

 with the terms of a single series (a y ) such that y<z. We 

 can then choose another number (of the second class) x x >x, 

 and we can correlate the terms of ( a Up) such that#<.a + /S<i?i 

 with the terms of (a v ) such that x<y<x x . We can continue 

 this process indefinitely ; and Mr. Jourdain infers from this 

 that we can correlate all the terms of the double series 



(aWp)^^ 1 with the terms of the single series (a y ) y<e>i. 

 The conclusion is true, but it does not follow from the 

 premisses f; al l that Mr. Jourdain's argument proves is that 



* It should be observed that we do not require any step by step 

 process of selection to obtain these aggregates, they are all completely 

 determined as soon as we define the correlation between M and N. 



t I may illustrate my argument by observing" that if we take a 

 progression we can correlate every alternate term of it with the ordinals 

 less than jSj and every alternate term of the remainder with the ordinals 



— ^ \ and so on (i3 1 </3 2 < ... <qj 1 ). We can continue this process 



