
Numbers of Well-ordered Aggregates. 67 
gates +; and, for a name for an aggregate which implies 
that the aggregate is consistent,—and, consequently, has a 
cardinal number, and, if (simply) ordered, an ordinal type,— 
I shall, in future, use the word manifold t. 
Cantor has repeatedly emphasized his view that what is 
essential in the conception of a manifold is the “ collection by 
the mind of definite, distinct objects to a whole” §. In con- 
formity with this view, we may attempt to define an in- 
consistent aggregate as an aggregate of which it is impossible 
to think as a whole without contradiction. 
But for formal purposes, I use the following definition : 
An ‘inconsistent’ aggregate is an aggregate such that there 
is a part of it which is equivalent to W. Further, since ‘‘ M 
is a consistent aggregate” is a necessary hypothesis (7. e., an 
hypothesis necessary to avoid contradiction) to the definitions 
of the cardinai number and type of M, we cannot use W (the 
ageregate of all ordinal numbers) in the above definition ; 
but must use the (well-ordered) aggregate of which every 
well-ordered aggregate is a segment ||, which is ordinally 
sunilar to W. : 

a 
If, now, a cardinal number could be greater than any 
Aleph, there must be a part of the aggregate to which this 
cardinal number belongs, which*can be made to have a 
one-one correspondence with W ; that is to say, the aggregate 
first mentioned is also inconsistent, and hence there ‘cannot 
be such a cardinal number. 
We may express this in other words by saying that the 
cardinal number of every manifold is a definite Aleph. Con- 
sequently all those manifolds whose cardinal numbers, although 
not primarily defined as Alephs, are known, must be Alephs. 
Such cardinal numbers are those of the continuum (0... 1) 
of real numbers, of real one-valued (or even {-valued) functions 
of one, or a finite number, or even No, real variables, and of 
all (even €-valued) functions of € real variables ; which are 
known to be respectively, 
c 
c=2*, 2° and 2?. 
+ The name I have taken from Schréder (‘ Vorlesungen iiber die 
Algebra der Logils,’i.). Cantor had also arrived, long before me, at the 
same concept and name; but I only learnt of these (unpublished) investi- 
gations of his after communicating my results to him. I shall shortly 
return to this point. 
t Thus, one may speak of the well-ordered aggregate (1), which has 
no ordinal number. 
§ Cf., e.g. ‘Grundlagen,’ p. 53, remark (1); Math. Ann. xlvi. p. 481. 
|| See Schonflies, op. cit., pp. 36, 40, 41. 
¥ 2 
