Numbers of Well-ordered Aggregates. id 
If & belongs to the class considered by Whitehead + in a 
series of interesting propositions, and which is characterized 
by the property that 
2 Se ee Gb) 
or, what is evidently the same thing, that there exists a 
cardinal number J such that 
the theorem F at once follows. For then 
€A=N,.€a=(N)4+&%))I=a +a; 
and 
a+b<a+a,a+b>a; 
so that 
er inti eh eid Hl 
Also—and this was not pointed out by Whitehead—ftrom 
(13) follows (12). For we can apply the conclusion from 
fy} to Np, of § 8 to the equality 
ee ipl — er kot ee. a ee 
which results from the hypothesis. 
If, now, M is a definite one of the well-ordered manifolds 
of cardinal number @, and we replace each element of M by 
a well-ordered manifold of two elements ; then, since M con- 
sists of a cardinal number 0 ¢ of series, each of which is of 
type w, together with perhaps, a finite number (v) of elements §, 
the manifold resulting from M also consists of BD series, each 
of which is of type , together with, perhaps, a finite number 
(2v) of elements. This results from the known equation 
Z.O=0. 
Since then, the resulting manifold is of cardinal number 
ad-+d, 
we get equation (14). 
Thus if & is any transfinite cardinal number, @ is unaltered 
by the addition of any (finite or transfinite) cardinal number 
DB equal to or less than @, and of these alone. Accordingly 
the class of such numbers D is hereby completely determined, 
and consequently for any cardinal number @ the following 
rules of calculation hold : 
+ Op. cit. p. 393. 
~ It is easy to see that b=a. 
§ Thus, if a=&,, M may be of type Q+o@+». 
