296 Mr. Jourdain on the Transjinite Cardinal 
and so on. I¢ is evident that the numbers of the first and 
second number-classes, arranged in order of magnitude, are 
ordinally similar to the series 
{@,+ 4}. 
We may then conclude, in a precisely similar way to that 
in which Cantor* has proved that any part of the second 
number-class has a cardinal number which is either finite, 
or Xo, or that of the second number-class, that any part of 
the third class has a cardinal number which is either finite, 
or No, or &,, or that of the whole third class. Thus, we have, 
and have only, to prove that this last cardinal number is not 
N:, in order to complete the proof that the cardinal number 
of the third class is the next greater one to &, and is thus 
properly denoted by No. 
For this purpose, we must prove f that if we supposed 
that all numbers of the third class could be arranged in a 
well-ordered series of cardinal number &,, we could define 
a number which belongs to the third class but not to this 
series. From this contradiction we must conclude that the 
cardinal number of the third class is not \; f. 
Suppose, then, that all the numbers of the third class 
could be arranged in a series of type @, (of course, not in 
order of magnitude), as would be possible if the third class 
were of cardinal number &,. Now any series (not necessarily 
arranged in order of magnitude) {8,{ of type ; of numbers 
of the third class has either a greatest number or else there 
is a number of the third class such that it is both the upper 
limit and a Limes § of 48,' when the latter is arranged in 
* “Grundlagen, pp. 37-88 ; Math. Ann. xlix. pp. 228-229. 
: + Cf. Cantor, ‘Grundlagen,’ pp. 35-36; Math. Ann. xlix. pp. 227- 
228. 
{ This method, which Cantor has used to prove that 
{, — 90. >No WS >No and ot >a; 
fails if we try to prove that £>,. Since, then, £7 &,, we have some 
grounds for believing that f=). 
§ It appears to me necessary to distinguish the two different concepts 
of limit by two different words such as limit and Limes. These con- 
ceptions are, as is evident from their use in the general theory of ordinal 
types, purely ordinal, but it is in the theory of real numbers and their 
functions that their application is most important. Any infinite manifold 
of real (or complex) numbers has at least one point of condensation 
(Weierstrass), and each of these points (which always exist, e. y., when 
the elements of the manifold are real numbers), which are the points of 
Cantor’s “first derivative,” is called a Limes. Especially important 
among the Limites are the greatest and least Limes (which always exist, 
