68 Mr. Jourdain on the Transfinite Cardinal 
b. 
We also deduce, from the known fact + of the magnitude 
relations of any two ordinal numbers, and, consequently, of 
any two Alephs, the theorem that: if & and D are any two 
cardinal numbers, then one of the relations: ‘less than,’ 
‘equal to,’ or ‘ greater than,’ holds between & and D. 
This theorem was stated by Cantor in 1895], but the 
proof was postponed until a view over the ascending sequence 
of the transfinite cardinal numbers and an insight into their 
connexion had been obtained. This is, now, supplied by the 
theorem that every cardinal number is an Aleph § ; but it 
may be observed that even if the general occurrence of one 
of the relations <, =, or > between any two cardinal numbers. 
& and B could be proved independently, it would by no means 
follow, inversely, that the cardinal numbers, when arranged 
in order of magnitude, form a well-ordered aggregate. For 
this would only prove that the cardinal numbers formed a 
simply-orderedaggregate. But further, even if we also knew 
that to every cardinal number was one immediately greater, 
and there was no greatest cardinal number, but there was a 
least, there would still be nothing to prevent the aggregate 
of all cardinal numbers from being, for example, of type 
a+*o+a. 
In this aggregate there is an immediate predecessor to every 
element, but in the aggregate of type 
o+*o+o.2, 
this is not the case. Still, neither of the aggregates is well- 
ordered, since in each there is a part of type 
Fay, 
i 
From the theorem: 
A. If @ and B are any two cardinal numbers, then either 
a=b, or &<D, or ADD. 
Jantor deduced || the following theorems : 
B. If two manifolds M and N are such that M is equivalent 
to a part N, of N, and N toa part M, of M, then M 
and N are equivalent ; 
+ Math. Ann. xix. p. 216. 
t Math. Ann. xlvi. p. 484. 
§ When, of course, the existence of the series of Alephs itself is clearly 
established, as T shall endeayour to do in a continuation of this paper | 
(cf. below, § 10). - || Ibid. foot of p. 484. 
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