
Numbers of Number-Classes in General. 297 
the (in general different) order of magnitude. This is proved 
in the following manner :— 
Let 8, be the first number in {@,? (arranged, as at first, 
2 
in a series of type @) which is greater than 8,, 8, the first 
*3 
greater than 8, and so on; so that there is determined in 
= 
{8,,} a series, of type @; at most, 
eee 8 <8 = eee 
“2 eee x Hh we. : 
— 
where also 
a5 ee. 2, i SS rae 
_ 
W 2 
Then, either there is a number 8. in (1) such that all the 
a 
numbers following it in {@,! are less than it,—in which 
case 6 is evidently the greatest number of { B,¢s,—or we 
employ the following argument. By completing the series 
(1) by intercalating, in order of magnitude, all the numbers 
{ry of the first three classes such that 
y < Bi, B <= iY < eh 2 
a a+l 
we get a definite part of the first three classes, which is at 
most of cardinal number 
Ni NL 
we can then prove the existence of an upper Limes, of the 
third class, of series (1), and consequently of a number of 
the third class not in {8,', if we can prove that 
Ni. NHN, 3 
we shall, then, have proved that the cardinal number of the 
third class is the next greater one to &,, when we have proved 
this equation. 
For this purpose, we shall extend the method for proving 
that 
X - NRo=NX 
by the diagonal enumeration of a double series to the case in 
which each single series is of type @, instead of type o. 

since the derivatives contain all their points of condensation). The 
(upper) Zimit, on the other hand, may be either the upper Limes or the 
sreatest value not a Limes, if such exists, of the manifold considered. 
hus, the two concepts are not in general coincident. The importance 
of the notion of a continuous function arises partly from the fact that 
the (upper) Limes is the (upper) limit, and inversely. 
