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XL. A (reneral Theorem on the Transjinite Cardinal JS^unihers 

 of Aqareoates of Functions. ^//Philip E. B. JouRDAm, 

 J3.A., Trinity College^ Canihridge^. 



THE (traiisfinite) ciirdiniil numbers o£ certain aggregates o£ 

 functions have been stated by Cantor j and Borel |. The 

 <luestion can now be treated much more shortly on the basis of 

 the rules of calculation for transfinite numbers introduced by 

 Cantor § and completed by Whitehead ||. It forms the most 

 general question that can be asked respecting aggregates of 

 relations between elements whose cardinal number alone is 

 given (i. e. in which we abstract from their ordinal types, if 

 these exist). 



In the following the letters H, &» U, t denote any finite or 

 transfinite cardinal numbers, and S has the same meaning 

 as it has with Cantor. 



Consider a function determined by the following deter- 

 minations : — 



Calling, for shortness, every system of values of t inde- 

 pendent variables a *' point " (of an arithmetic space If of 

 dimensions, Rt); suppose that to each such point belong 

 b values of a function, and each of these values can be chosen 

 from an aggregate of cardinal number H. Then to each point 

 of R^ case belongs an aggregate of cardinal number 



a». 



Further, suppose that each variable need only take an 

 aggregate of cardinal number of values in order that the 

 function be completely determined for the whole of Re. Then 

 the cardinal number of all the " argument-points ''' required is 



Consequently, the cardinal number of all the functions 

 considered is 



(a*)"' (1) 



This general formula may now be applied to the cases 

 hitherto know^n. 



* Communicated by the Author. 

 t Math. Ann. xxi. p. 590 (1883). 



X Legons sur la Theorie des Fonctions, pp. 125, 126 (1898). 

 § Math. Ann. xlvi. (1895). 

 il Amer. Journ. of Math. xiv. (1902). 



51 We do not .suppose tliat the domain of each variable in R^ is neces- 

 sai'ily of the cardinal number c of the continuum. 



Y2 



y .-: 



