Numbers of Well-ordered Aggregates. 63 
(2), like o, .2, w.v, w®, ©, ...), N, is the next greater 
cardinal number to all the cardinal numbers Nz such that 
p<y. 
Further, Np is the smallest transfinite cardinal number fF ; 
it is the cardinal number of any enumerable ageregate, an 
aggregate which can always be re-arranged (if a re-art “ange- 
ment: is necessary) in the form of a well-ordered ag oregute 
of type o. 
ih: 
We have thus an illimitable series of ascending cardinal 
numbers. But there are cardinal numbers, such as the 
cardinal number C of the real-number-continuum (0 ....1), 
which are not defined as cardinal numbers of well-ordered 
aggregates, and of which we cannot therefore immediately 
say that they occur in the series (1). However, Cantor 
showed tf that 
ao Ee ee ee a eee, 
and has always believed § that 
C=, 3 
though the latter equality has never yet been proved. But 
since N, is the next cardinal number to N, it is possible to 
take elements from the number-continuum corresponding to all 
the numbers of Cantor’s first and second classes of ordinal 
numbers. For if this process were to stop we should have 
C=), 
which is contradicted by (3). Using, then, the theorem of 
Schréder and Bernstein ||, we can state that 
EAN §- 
If, now (>, we can conclude similarly that 
C22; 
and soon. And if 
C>X, 
+ Math. Ann, xlvi. p. 492. (Cf. Cantor, “Ein Beitrag zur Mannig- 
faltigkeitslehre,” Journ. fiir Math. \xxxiv. 1878, p- 242.) 
T ‘Ueber eine Eigenschaft des Inbegriffs aller reellen algebraischen 
Zahlen,” Journ. fiir Math. \xxvii. 1874, pp. 258-263. 
§ Cf. “Ein Beitrag zur MonmniefaltieWerislalir,” Journ. fiir Math. 
lxxxiv. 1878, p. 257 ; Grundlagen,” pf. 
|| See below, §§ 7 and 8. 
q The fact that 2o>~N, follows, perhaps even more simply, from the 
consideration that 2 2%, is the cardinal number of the transfinite classes 
which can be formed out of &, members, while &, is the cardinal 
number of some of these classes. 
