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III, On Transfinite Cardinal Numbers of the Exponential 

 Form. By Philip E. B. Jourdain, B.A., Trinity College, 

 Cambridge *. 



MONGr cardinal numbers of the form 



A 1 



where u , at least, is transfinite, the smallest and most 

 interesting is the cardinal number of the number-continuum : 



Cantor has always been of the conviction that 



and investigations in the theory of manifolds tend to increase 

 one's belief in the truth of this conviction, although hitherto 

 no proof of it has been given. It is very important to prove 



that 2**° is equal to some Aleph in order to be certain that 

 the number-continuum is not what I have called an 

 "inconsistent aggregate'" f. 



A failure to prove the above equality by an attempted 

 arrangement of all the real numbers between and 1 in a 

 well-ordered series ultimately led me to the result of § 1, 

 that the cardinal number of all the real numbers which can 

 be represented by fundamental series of which the general 

 term is known as a rational function of its index is tt , which 

 proves that it is impossible to obtain a series of type o^ from 

 such numbers, and consequently the impossibility of actually 

 proving that 



in a large class of cases. 



This negative result, which is the only definite result 

 I have as yet been able to obtain on the question of the 

 equality 



where a is any ordinal number, allows, however, a number of 

 conclusions to be drawn in what I have called the " cardinal 

 theory of functions'" (§2). The result that only a small 



* Communicated by the Author. 



t Phil. Mag. January 1904, p. 66. In § 5 (p. 67) of this article I 

 tacitly assumed that the exponential numbers in question belonged to 

 consistent aggregates, or manifolds ; for, though this is not rigorously 

 proved to be the case, nothing seems more unlikely than that it should 

 not be so. Further information on the subject of inconsistent aggregates 

 is given below, §§ 6-9. 



