46 Mr. Jourdain on Tramfinite Cardinal 



integrable, is not. in general, representable as. the limit of a 

 sequence of continuous functions*. 



This example suffices to substantiate the contention that 

 the requirement of arithmetical definability is unnecessarily 

 narrow for the possibility of a general theory of functions. 

 In other words, there exist propositions in the general theory 

 of functions (on integrable functions, for example) which 

 apply to a much wider class of functions than that of 

 arithmetically definable functions. 



Now the class of functions which can be represented as 

 limits of infinite series of continuous functions, or, what is 

 the same thing, of functions to which an " existence-theorem " 

 is applicable, contains, of course, all arithmetically definable 

 functions, but not inversely. For every function of the 

 former class is completely and uniquely determined by the 

 datum of the enumerable sequence of the coefficients of 

 the sequence of polynomials by which it can be replaced, and 

 the cardinal number of the sequences (1) whose general term 

 can be found in the manner indicated, but in which, possibly, 

 a finite number of terms are completely arbitrary, is 



and v , 



3. 



Although there is thus no possibility of actually constructing 

 a greater cardinal number of functions than X , it by no 

 means follows that definite theorems cannot be found which 

 hold for a greater number. The fact of the existence of a 

 general theory of analytic functions is alone sufficient to 

 disprove this, and, consequently, also for this reason the 

 requirement of the arithmetical definability of functions is 

 too narrow. 



Further, it is interesting to see that there is a theorem 

 which holds of actually non-representable analytic functions, 

 due to Borel and Fabry f. The series 



a + a 1 z + a 2 z 2 + ... + a v z v + ..., . ... (2) 



where z is a complex variable, represents either the whole of 

 an analytic function or part of one within a circle on whose 

 circumference is at least one singularity. The theorem of 



* Messenger of Math. Sept 1903. 



f Cf. Hadamard, ' La serie de Taylor et son prolongement analytique,' 

 Paris, 1901, pp. 33-36. 



