72 Mr. Jourdain on the Transfinite Cardinal 
cardinal number @. Proceeding in this way, we obtain a 
series of manifolds 
Py Pay Pay cacy Pos ooh tee 
and this series can stop at no finite v, for, if it did, the 
equations (7) would be contradicted. Further, each P, has 
no point in common with any other P, and there exists fT a 
manifold 
My» 
which may, however, consist of no elements, which is the 
jirst (in the above process) of all manifolds which is not 
contained in all the manifolds P,. ‘This Mw is thus defined 
by essentially ordinal considerations. 
Let, then, & be the cardinal number of M, ; then, since 
the series (11) is of type w, and therefore of cardinal number 
No, while each P, is of cardinal number (10) ; we have 
a=wn,(b+e)+9 
=2.x,(0+e)+9=[%,(b +e) +9]+8,(b +e) 
—=A8+,(b+e), 
which is the required equation (8). 
Accordingly, the independent proof of the theorem B 
appears to depend essentially on ordinal conceptions, although 
it is true { that whatever may be the cardinal numbers of 
M and N, the proof requires only an enwmerable manifold 
(of type w +1) of steps. 
oi. 
We may now deduce some general laws of calculation with 
transfinite cardinal numbers, analogous to those given by 
Cantor § for No; namely, where v is any finite cardinal 
number, 
No tv=No, v No=No, No =N- 
For this purpose I shall now prove the first of the two 
theorems denoted by Whitehead || in his memoir “On 
Cardinal Numbers” as unsolved, namely : 
If & and D are cardinal vaca: @ is transfinite, and 
ab; 
then | 
F. a+b=a; 
Cf. Schonflies, op. ct. p. 14. 
t Cf Schonflies, op. cit. p. 18. 
§ Math. Ann. xlvi. pp. 492-495. 
Amer. Journ. of Math, xxiv. pp. 367-894 (1902) ; see especially, 
pp. 368, 381-383, 393. 
