74. Transfinite Cardinal Numbers of Well-ordered Aggregates. 
ai+b=a (b=a), 
a+). b=a, 
‘ - @=Np.d=X, a: 
The first two equations contain the properties proved by 
Zermelo for the class of numbers BD, which, however, was not 
determined by him. This class of numbers was called by 
him a “group belonging to &” inasmuch as the members 
reproduce themselves or other members of the class by their 
diminution, by their multiplication with N,, and by their 
addition in a finite or enumerable manifold of summands. 
If we add that the members also reproduce other members of 
the class also by ¢ncrease, when this is necessary, till they 
become equal to &, we have a characterization of the group 
in question. Further, we shall prove later that 
=a, 
a+b’=a.b'=a, 
which gives a further self-reproductive property cf members 
of the group. | 
and so that 
LG: 
If we attempt to use the method of § 9 to prove the second 
theorem denoted by Whitehead + as unproved; namely, 
a.b=a (Da); 
= 8s. 5 J. er 
we are met by the fact that, just as it is necessary to have 
proved that 
or its equivalent 
= 
in order to prove { that 
so it is necessary to prove previously the equation (15), or 
2 
NY=N,, 
where y is any ordinal number, before the existence of the 
series of Alephs (1) can be proved. It is, then, necessary to 
investigate in greater detail the series (1) {. The importance 
of this may be considered as established by the fact that, 
having arrived at (1), we are sure that every transfinite 
+ Op. cit. p. 868, Whitehead remarked that (15) does not follow from 
12). 
t See ‘Grundlagen,’ pp. 35-36; Math. Ann. xlix. pp. 227-228, 222. 
§ Cantor has hitherto only treated in detail the ordinal number of the 
first two classes and the cardinal numbers of these classes. 
