Numbers of Well-ordered Aggregates. 69 
C. If M, is a part of M, M, is a part of M,, and Mand M, 
are equivalent, then M, is equivalent to M ; 
D. If N is equivalent neither to M nor toa part of M, 
then there is a part of N which is equivalent to M : 
K. If Mand N are not equivalent, and if there is a part 
of N which is equivalent to M, then no part of M is 
equivalent to N. 
Of these, B has been proved independently by Schroder fF 
and by Bernstein f, and hence, if @ and BD are the cardinal 
numbers of, respectively, M and N, and M is equivalent to 
some part of N, we can always say that 
a<b. 
The theorems C and HK, therefore, follow from A, and are 
thus proved independently of the theorem that every cardinal 
number is an Aleph. 
As for the theorem D, it is easy to see that it both implies 
and is implied by the theorem : 
Itis impossible that neither a part ot M should be equivalent 
to N nor a part of N should be equivalent to M, provided 
that both M and N are transfinite. 
This theorem contains the settlement of the fourth possi- 
bility in the relations of equivalence of M and N and parts 
M,, N, of them, which is the only one left undecided by the 
definitions of 
a<b,a>b, 
and the theorem of Schréder and Bernstein §. 
In the suppositions which I have made,—namely, in the 
proof of the theorem that every cardinal number is either an 
Aleph or is greater than any Aleph—this theorem of Schréder 
and Bernstein is already used, so that D is the only one of 
the theorems B to E which can, without a circle, be deduced 
from my proof that every cardinal number is an Aleph. 

t “Ueber G. Cantor’sche Siitze,” Jahresber d. d. Math.-Ver. v. pp. 81- 
82 (1897); “‘ Ueber zwei Definitionen der Endlichkeit und G. Cantor’sche 
Satze,” Nova Acta Leop.-Carol. Akad. (Halle) 1xxi. pp. 303-362 (1898). 
{ Borel, “Legons sur la théorie des fonctions,” Paris, 1898, pp. 104— 
107. Cf. Schontflies, op. cit. pp. 16-18. 
§ See Borel, op. cit. p. 102; Schénflies, op. cit. p. 15. It may be 
remarked that the statement in Borel’s book, p. 103, may lead one to 
error. In his fourth case, it is obviously impossible that A should be 
equivalent to B if A and B are transfinite (for if it were, A would also 
be equivalent to some part of B) and Borel apparently contemplates this 
impossible state of things as possible and says that if, in this fourth case, 
A is not equivalent to B, there can be aggregates A and B such that 
; © ; 
their cardinal numbers are not comparable in respect of magnitude, 
