44 Mr. Jourdain on Tmnsfinite Cardinal 



the real numbers which can now be represented by the 

 sequence (1) is merely K . This may be shown as follows. 



The function obtained by performing the elementary 

 operations a finite number of times on (n + 1) arguments is a 

 rational function of arguments, the coefficients of which are 

 integers *. Since, then, in each case we only have a finite 

 number (m) of coefficients to choose, and each coefficient can 

 be chosen out of N values (the integers), the cardinal number 

 of those functions of m coefficients is 





 Further, we get all such functions by giving m all possible 



finite values in turn ; consequently the cardinal number of 

 all these functions is 



N 1 +N 2 +...+*C + ..., 

 the series being of type co, and consequently — remembering 

 that each term reduces to tf — the cardinal number in 

 question is 



Nc.No-No- 

 We may state this result in words as follows : The cardinal 

 number of all the real numbers that we can actually determine 

 (that is to say, determine in the sense explained above) is 



Mo- 



Accordingly, if, as is the case with some methods that 

 suggest themselves for arranging real numbers in a well- 

 ordered series, we only use such "actually determinable * 

 real numbers, we can never arrange them in a series of 

 type ewj. For every enumerable manifold can be well-ordered, 

 but the series always breaks off before some number of the 

 second number-class is reached. 



Now, this conclusion has applications, which seem to me 

 to be of some importance, in the theory of functions. In the 

 first place, such sequences as (1) enter into Weierstrass' 

 construction of whole transcendental functions with given 

 zeros, Mittag-Leffler's construction of analytic functions 

 whose singularities form an aggregate whose first derivative 

 is enumerable, and the construction of whole transcendental 

 functions which take given values at certain points f. We 



* Cf. Harnack, ''An Introduction to the Elements of the Differential 

 and Integral Calculus/' Eng. trans, p. 67 (1891 ). 



f This construction, which forms an extension of Lagrange's inter- 

 polation-formula to whole transcendental functions, is given by me in 

 part of an essay " On the General Theory of Functions/' which is to 

 appear shortly in Crelle's Journal fur Math. It is very simple, and is 

 obtained by the multiplication of a whole function constructed by 

 Weierstrass' theorem with a meromorphic function constructed by 

 Mittag-Leffler's theorem. 



