Numbers of Well-ordered Aggregates. ut 
Hence | i 
BEA VE ee 
where v is any finite cardinal number. 
Tf now, we can conclude, from (7), that also 
a=atXQi(bt+e), - - --- (8% 
we can say, because the right-hand side of (7) becomes 
at®,.b+ (% 4+ De=[ats(b+e)]+e, 
that 
a=arte. 
me eee es mere oes ED) 
From (5) and (9), now, 
D=4, 
Similarly 
or 
at+b=0+e. 
which is the equation required. 
This part of the proof is purely cardinal; the ordinal part 
appears in the proof of equation (8) from equations (7) ; in 
other words, in the proof that it is possible to conclude from 
4v¢ to NT in the case of (7). 
From the manifold M,, of cardinal number @, we can take 
away, by (6), a manifold (P,) of cardinal number 
ee ee ak i 
while the remainder has still the cardinal number @ ; let M, 
be this remaining manifold. From M, we can again (by (7)) 
take in a similar manner, a manifold (P.) of the cardinal 
number (10), and we thus obtain another remainder M, of 
+ That one cannot, in general, conclude from }vf{ to &, is evident from 
the consideration of the known relations : 
NU=N N= Sy. 
That, however, this “ extended principle of cardinal induction” is legiti- 
mate in the case of the text, I had found, independently of Zermelo, in 
October, 1902; and the remark of the ordinal character of the proof has 
led me to emphasize this point in Zermelo’s proof. 
The extended principle of ordinal induction, or conclusion from {vr} 
to w, as used by Schonflies (op. cit. pp. 45, 52, 60, 67, 235), should be 
compared with this. It seems true that mathematics is principally 
occupied with sufficient (and necessary and sufficient, in closer investi- 
gations) conditions under which one can conclude from {v} tow. Thus, 
if {s,(x)} bea convergent sequence of functions, and s,,(¢) thus properly 
denotes its limit, the most fundamental problem here is to know when 
one can conclude from {s,(x)} to s (x) as to continuity, regularity, etc., 
and uniformity of convergence is important because it gives a wide 
sufficient condition. 
