Transfinite Cardinal Numbers of Exponential Form. 43 



portion of the whole manifold of analytic functions, for 

 example, are analytically representable by no means implies 

 that general theorems cannot be found which apply to all 

 analytic functions, and even in particular to all which are 

 not representable (§ 3) ; so that the concept of function taken 

 by Pringsheim, in the recent Encyclopadie der mathematiscJien 

 Wissenschaften*) as the basis of the general theory of 

 functions, appears to be too narrow. 



After a digression on the cardinal theory of functions and 

 on the utility of the concept of the u aggregate of definition" 

 (§4). I prove (§ 5) a theorem due to Bernstein on exponential 

 numbers, which includes a result of my own f as a special 

 case, and allows us to rind the necessary and sufficient 

 conditions that 



where cl and b are any cardinal numbers. 



In § (5. I make a few remarks on the extended principle 

 of induction used in § 5, which serves to define the series W 

 of ordinal numbers. The series (2ZI) such that every well- 

 ordered series is ordinally similar either to 221 or to a 

 segment of 2£l extends beyond W (§7), and this more 

 exact account of TV throws a clearer light on my solution of 

 Burali-Forti^s contradiction (§8). 



Finally, in § 9, I revert to the consideration of the concept 

 of '•consistency/' with especial reference to investigations of 

 Cantor, Hilbert, and Russell. 



1. 



Every real number is determined by an enumerable sequence 

 of rational numbers, and hence the cardinal number of the 



aggregate of real numbers is seen without difficult v to be 2 kN '". 

 But, if this enumerable sequence is 



Mli «2J ■■> u n ••> (1) 



we must, if we are to be able to determine exactly the real 

 number in question, limit the form of u v to be a function 

 obtained by performing the elementary operations a finite 

 number of times on v and a finite number (n) of given 

 rational numbers ; in symbols 



"v=f(y, i'\, r 2} ..., >•„). 

 By this limitation, the cardinal number of the aggregate of 



* Bd. ii. A. 1, pp. 0-11. 



t Phil Mag. March 1904* p. 302. 



