70 Mr. Jourdain on the Transjinte Cardinal 
However, Professor Uantor t drew my attention to the fact 
that the method of the proof does not require this supposition. 
In fact, we can, successively, make the elements of W corre- 
spond to elements (in any order) of any given aggregate C. 
If this process comes to an end, the cardinal number of C is an 
Aleph; if not,C must contain as part the inconsistent aggregate 
W, and is thus itself inconsistent and has no cardinal number. 
With this formulation we obtain a new proof of the theorem 
proved by Schréder and Bernstein. 
The problem as to the relations of magnitude of any two 
cardinal numbers is thus completely solved by the consider- 
ation of the well-ordered aggregate of the cardinal numbers 
of well-ordered manifolds. This entry of ordinal notions 
has not appeared satisfactory to Schréder f, since the question 
is one of the elementary properties of cardinal numbers ; 
but, in fact, Schréder’s proof also involves ordinal concep- 
tions. To show this, I have given, in preference to giving 
a version of the original proof, a slightly different form to 
Zermelo’s§ proof; for this latter proof may be described 
as an exceedingly happy analysis of the Schréder-Bernstein 
preof,—analysis because the ordinal conceptions are brought 
out more clearly in it. 

8. 
Let a, D, DU. and & be any four cardinal numbers, such that 
a=D+e,0=a-b .-. ) oa 
This is the statement, in the language of cardinal numbers, 
of the hypothesis in Cantor’s theorem B (which is that 
proved by Schroder and Bernstein), when the cardinal 
numbers of M,, N,, M-M,, N-N, are respectively @, b, b, 
and &. 
Then we have to prove that 
aib=D-+e. 
From (5), 
D+ie=‘at+b) +e=a+(b+e) ;sx 
or 
+ Ina letter of November 4th, 1903. I had previously communicated to 
Professor Cantor my proof that every cardinal number is an Aleph, and, 
in this reply, he gave the (unpublished) proof in essentials identical with 
mine, which he had used in 1895 to establish the theorem A of §7, and had 
communicated in 1896 to Professor Hilbert and in 1899 to Professor 
Dedekind. I am indebted to Professor Cantor for his kindly encourage- 
ment to publish my proof. 
t Nova Acta Leop.-Carol. Akad. \xxi. p. 303 (1898). 
§ ‘Ueber die Addition transfiniter Cardinalzahlen,” Gott. Nachr., 1901, 
pp. 34-38. 
