Numbers of Number- Classes in General. 301 
The equation (6) is, then, quite general; and, since every 
transfinite cardinal number is an Aleph, we can_ state 
generally: If& be any transfinite cardinal number, then 
a=a=a’, 
where v is any finite cardinal number; or, if B is any cardinal 
number such that 
b<a, 
then 
u D=a. «2; eee 10) 
The equation (7) gives a new property of the numbers 
forming the A-group described in § 10 of my former paper. 
3. 
From the equation (7) we may also conclude that that 
class of type @, and, consequently, of cardinal number N,, 
consists, if y has an immediate predecessor y—1, of Ny 
consecutive series, each of which is of type w,_, and cardinal 
number N,_,. For the class w, is, of course, built up from 
series of type w,_, and the cardinal number of these latter 
series is, by (7), greater than 8, _, and less than X,,). 
When ¥ is a Limes-number, the number-class of type Oo, 
is built up of series 3, arranged in the order of the 
suffixes 5, where {6} is any ascending fundamental series 
when this term is extended to denote series whose type is 
other than @ ; say ®@+¥, @;, or w,) such that its upper limit 
isa Limes, y. We would have arrived at the same class w, 
if any other series {6} of upper limit-Limes y had been 
taken. 
4. 
From the theorems 
2,a=a and a=, 
we can obtain some other general theorems on transfinite 
cardinal numbers. 
Since there is a part of a manifold with the cardinal 
number 
qa 
which has a one-one correspondence with a manifold of 
eardinal number 
ree 
Phil. Mag. 8. 6. Vol. 7. No, 39. March 1904. ay 
