
Numbers of Number-Classes in General. . 295 
by proving (§1) that the cardinal number of the third 
number-class (N2) is the next greater cardinal number to N;. 
Cantor has already done this for &; and Np, and has not done 
this in detail for any other Alephs ; though there are only 
the generalizations introduced by having to deal with series 
of the type of the second number-class instead of series of 
type w. The proof depends on the fact that 
NS; : Ni =N), 
which is proved directly. There is no great difficulty in 
advancing to &,, &,, and to any Ny *, so that the theorem 
a? =a 
is proved for any cardinal number @. 
This gives certain information as to the constitution of any 
number-class + (§ 3). According as the corresponding 
cardinal number has or has not an immediate predecessor, 
the class is built up in one of two ways from the lower 
number-classes. 
Finally, such equations as 
qa—a 
are proved, by the results in the addition and multiplication 
of transfinite cardinal numbers which have been hitherto 
obtained, to hold if & is any transfinite cardinal number (¢ 4). 
1. 
If we denote by a, the first number of the third number- 
class, the whole class is formed as follows :—First after o, 
comes the series represented by 
f@,+ a}, 
where a takes, in order of magnitude, the values of all the 
numbers of the first and second number-classes. Next after 
this series comes the number 
@, -- @,—@,. ye 
which is followed by the series 
$@ E 2 +a} 2 
* A doubt as to whether Cantor’s second principle of generation can 
lead to numbers y beyond the second number-class, which seems to have 
been held by Schénflies, is, I think, settled below (§ 2). 
7 The notation w, is used for the first ordinal number of the (y+2)th 
class. This notation (which is borrowed from Russell, cf. ‘The Principles 
of Mathematics,’ Cambridge, 1903, p. 322) is necessary when we deal 
generally with the yth number class. Cantor has only hitherto had to 
use a notation for what is here called w, : he wrote Q. 
