Numbers of the Exponential Form. 45 



conclude then that, although the cardinal number o£ any of 

 the above classes of functions is 2 °, the cardinal number 

 of those functions which we can actually represent is K . 

 Of course, we make the same stipulation as to representability 

 in the case of the extraneous factor in all these constructions. 

 In Weierstrass' construction this factor is 



*«, 



where g{z) is any whole function. Thus, we cannot, for 

 example, consider 



e.P(s) 



as a constructible function, if l?(z) is the product of primary 

 factors and c is any real number ; for c must be a representable 

 real number. 



In the second place, it appears that the postulate of 

 '' arithmetical definability/' which Pringsheim has introduced 

 as an essential qualification of the functions which can be 

 treated in a general theory of functions, cannot be considered 

 as relevant, for the double reason that it is necessary to take 

 account of functions which cannot be defined by tt conditions 

 and that even functions which are so definable are not in 

 general " arithmetically representable/' The former reason 

 rests on a theorem which constitutes an important part of 

 what I have called " the cardinal theory of functions " ; the 

 latter reason rests on a theorem which is easily obtained from 

 what precedes and completes, in a sense, the cardinal theory 

 of functions. 



2. 



The cardinal theory of functions consists of two parts : 

 The determination of the cardinal numbers of the various 

 aggregates of functions, and the drawing of conclusions, 

 from inequalities between these numbers, as to the non- 

 inclusion of certain aggregates in certain others. Thus, 

 from the results that the cardinal number of all integrable 

 functions is 



while that of all functions representable as limits of sequences 

 of continuous function- is 



and z ' 



we conclude that a function, even when it is restricted to be 



