66 Mr. Jourdain on the Transjinite Cardinal 
agoregate M should be well-ordered, it is necessary and 
sufficient + that M should contain no part of type 
*e, 
It is, now, easy to prove that the aggregate (2) is well- 
ordered {. If, namely, the aggregate of all ordinal numbers, 
arranged in order of magnitude, so that it is certainly a 
simply-ordered aggregate, were not well-ordered, there would 
be a part in it of type 
w 
let this be 
hy > Op > Sane aie ees ey (4) 
where 2, a, ..., 4, ... are ordinal numbers. Then in the 
well-ordered aggregate, A,, of all the ordinal numbers up to 
and including a, there would be, since all the numbers a,, 
G2, ..., %, ... would be contained in this aggregate, the partial 
agoregate (4) of type 
@; 
but it is impossible that a part of this type should be con- 
tained in the well-ordered aggregate Ay. 
Hence the aggregate (2), which I will call W, of all 
(=). o)y tS) 
ordinal numbers, arranged in order of magnitude, is well- 
ordered ; together with the aggregate (1) of all Alephs, which 
is similarly ordered to W. 
4, 
There arises, then, an insuperable contradiction if we 
speak of W, or any similar aggregate, as having a cardinal 
number or ordinal type; and, if the foundations of what is 
called the theory of aggregates, and with it the whole of pure 
mathematics §, is to be free from self-contradiction, we must 
agree that certain aggregates (like W) have no cardinal 
number and no ordinal type. It appears to me that such 
aggregates may be conveniently called inconsistent aggre- 
+ This characterization of well-ordered aggregates makes it very easy 
to construct simply-ordered aggregates which, in spite of their having 
many of the properties of well-ordered aggregates, are not well-ordered. 
Thus, an aggregate with a first term and an immediately consecutive 
term to every term in it (and even with an immediate predecessor to 
every term) need not be well-ordered (cf. end of §6). Such examples 
do not seem superfluous in view of the incorrect definition of a well- 
ordered aggregate given, e. g. by Hadamard (Verh. d. math. Congr. 
Ziirich, 1897). 
t Cf. Schonflies, op. crt. p. 41. 
§ To have given precision to the somewhat vague term “pure mathe- 
matics,” and to have shown that all pure mathematics depends uniquely 
on the logical concepts at the foundation of Cantor’s theory of aggregates, 
are two of the great merits of Russell, to whose work I have already 
referred. 

