48 Mr. Jourdain on Transfinite Cardinal 



Now, these theorems, together with an extension, follow at 

 once from a consideration of the character of the aggregates 

 of definition. 



Suppose that a one-valued analytic function, /(*), with a 

 period smaller in absolute amount than any real positive 

 number without being a constant exists, and \etf[z) have a 

 finite value for every point z such that |2J<a, where a is 

 some positive constant. Then there must be a sequence of 

 points {z v } condensing at some point z within the circle and 

 such that \z v \< a, such that 



/(*„)=/(*)= A. 



Now {z v } forms an aggregate of definition, and consequently 



/(--)- A. 



Further, if x and F(.i') are real, and F(x) is one-valued 

 and continuous, and ¥(x) has a period of the above nature, 

 it is easy to see that the points where F(x) is equal to some 

 number A lie everywhere dense, and thus form an aggregate 

 of definition of the continuous function. But it is evident 

 that this argument applies also to the case where x and 

 F(x) are complex and the periodicity of F(x) is double. 

 Hence, a real or complex function of a real or complex 

 variable cannot have a (respectively single or double) period 

 smaller in absolute amount than any positive non-zero number 

 provided only that the function is continuous. 



I now return to the consideration of exponential numbers 

 in general, and prove the theorem of Bernstein * that, if tt a 

 and N/3 are any two Alephs, 



«£'=*.. 2"' (3) 



In the first part f of Bernstein's proof, (3) is proved if 



* " Untersuchungen aus der Mengenlehre," Gott. Diss., Halle-a.-S., 

 1901, pp. 49-50. 



f A more general theorem than that of the^Vs^ part was proved by 

 me before I had seen Bernstein's memoir (Phil. Mag. March 1904, 



p. 302), namely: if K a <2**0 then tf^=2^. 



I take this opportunity of correcting two slips in this paper : 

 p. 303, line 7; delete u <2 a ." 



line 18 ; for " K^°= Ki " ], ™ d " N^° =2 kS V' 



