Numbers of the Exponential Form. 49 



and in the second parr it is supposed that 



Ka>tfs: 



and an extended form of complete induction, which extends 



to all Alephs. is used, and is. in essentials, as follows. 

 By Cantor's definition of an exponential number, 



is the cardinal number of all coverings (Belegungen) of a 

 manifold of cardinal number tt 8 with the elements of a mani- 

 fold of cardinal number tt a . which we will suppose to be 

 arranged in a series of type a) a . Xow every such covering 

 i- obtained by the covering of the manifold of cardinal 

 number Ns with some (or all) elements of some segment of 

 the above series of type o) a : and the cardinal number of this 

 segment is less than K z . Hence, each of the coverings first- 

 named is found among the aggregate of all the coverings of 

 the manifold of cardinal number Ns with the elements of the 

 various segments, taken in turn, of the series of type o> a . 

 Xow. the cardinal number of all the coverings of a manifold 

 of cardinal number X 8 with the elements of a segment M y of 

 the above series M s of type co a is 



Itl^X (4) 



where Hlis the cardinal number of M. Hence we can 

 state, by the Schroder-Bernstein theorem, that 



N^ < 2 («£'), (5) 



where the ^ means that the summation is to be extended 

 over all the numbers (4) such that M i- a segment of ]1 

 (or such that r y<w a ). 

 On the other hand. 



and hence the right-hand side of (5) is less than or equal to 

 Comparing this result and (5), we conclude that 



k>= 2 (m^) (6) 



v<w„ 



Hence, if we know a theorem for all Alephs less than K a , 

 we may, by substitution in the right-hand side of (6). find 

 PJtil. Mag. S. 6. Vol. lb No. 49. Jan\ 19C5. E 



