Numbers of Well-ordered Aagregates. 65 
who concluded from it that one must deny both Cantor’s T 
fundamental theorem in the theory of ordinal numbers that : 
if 2, and a, are any two ordinal numbers, then either 
4) <a, OF &,= ag, OF 4, > HQ; 
and the corresponding theorem for Alephs. This conclusion 
is, in fact, necessary if one admits Burali-Forti’s premisses ; 
and, since Cantor’s demonstration of the above theorem is 
beyond all pessible objection, Russell { avoided the contra- 
diction by denying the premiss that the series of all ordinal 
numbers, arranged in order of magnitude, is well-ordered. 
Then the ordinal type (@) of (1) or (2%) ceases to be an 
ordinal number, and we can no longer assert, in general, that 
B+1>8. 
Further, the cardinal number ceases to be necessarily an 
Aleph, and we cannot therefore assert, as we could before, 
that one of the Alephs surpasses it. 
But it appears possible to prove that this series of all 
ordinal numbers is well-ordered ; to this proof the following 
section is devoted. 
3) 
If we adopt, as definition of a well-ordered aggregate (M) 
the property which Cantor § has proved to be the charac- 
teristic of well-ordered aggregates among all simply-ordered 
aggregates ; namely, that both it and everyone of its partial 
aggregates should have a first element ; it becomes evident 
that no part of M can be of ordinal type 
*@ 
s 
But in no publication known to me does it appear to have 
been remarked explicitly that this gives a sufficient, as well as 
a necessary, condition that the simply-ordered aggregate M 
should be well-ordered. However, if M contains no part of 
ordinal type 
2 
@, 
it is well-ordered ; for if not, at least one of its parts would 
have no first element, and in this part a part of type 
*w 
can always be found. Thus, in order that a simply-ordered 
+ Math. Ann. xlix. p. 216. 
t ‘The Principles of Mathematics,’ Cambridge, 1903, p. 323. Cf. 
Cantor, Math. Ann. xlix. foot of p. 216. 
§ Math. Ann. xlix. pp. 208-209. This property has been taken as 
the definition of a well-ordered aggregate by Schonflies (op. cit. p. 36) 
and Russell (op. cit. p. 319, last note). 
Phil. Mag. 8. 6. Vol. 7. No. 37. Jan. 1904. Fr 
