50 Mr. Jourdain on Transfinite Cardinal 



the same theorem for K a . Such a theorem is 



where 



xA ■■ 



and (6) allows us to conclude the validity of the theorem (3), 

 even if 



From (3), we may deduce an interesting theorem concerning 

 those cardinal numbers which are unaltered by exponentiation. 

 In fact, from (3) and the laws of multiplication of Alephs *, 

 it follows that if, and only if 



■2^<a a . . • (7) 



the right-hand side of (3) reduces to X„. 

 Thus, that 



«^ = N« (8) 



it is necessary and sufficient that (7) should hold. In 

 particular, if, as is probable, we can assert (8) if only 



it is necessary and sufficient that 



In the extended principle of induction used above, which 

 may be stated thus : If a certain proposition P holds of tf„, 

 and if, when it holds of all Alephs less than X a , it holds of 

 tf a , P holds of all Alephs ; the proof of P for tt a is reduced, 

 by (6), to the proof of P for a sum (of cardinal number tf a ) 

 of numbers for which P is assumed to hold. This method 

 cannot be applied to give a shorter proof of the equality f 



since we must have previously proved that 



in order to prove that the cardinal number of a series of type 

 &) y +i is greater than that of a series of type (o y *. But if the 

 exponent, instead of being v, is transfinite, Ave can, as we 



* Phil. Mag. March 1904, p. 301. 



t Ibid. p. 300. This theorem seems, from an indication given by 

 Bernstein {op. cit. p. 49), to have been known to Cantor. 

 % Ibid. Jan. 1904, p. 74. 



