300 Mr. Jourdain on the Transfinite Cardinal 
Hence 
SI Ni =N) 3 
and the process can evidently be generalized so as to prove, 
by complete induction, that 
No ENy Le 
where v is any finite ordinal number. 
2. 
We have next to show that 
No NSN oot. 
Now, Nw is the cardinal number of that well-ordered 
manifold whose type is @, ; that is to say, of the series 
1, 2, nee Ys cee @, OF 1%... 8, 6. +o, (He@oy eee 
The extension of the method of §1, in which extension 
the equations (4) are used, to this case is obvious, and the 
equation (5) follows. But now we are certain that the 
equation 
N,.N,=8,, .. - - ee 
where y is any ordinal number, holds; for the extended 
method of diagonal enumeration of a double series allows us 
to conclude, from equations (like (4)) holding for the Alephs 
of a series of ordinal numbers, to an equation (like (5)) 
holding for the Limes-number of these ordinal numbers. 
But, that the equation (6) may be seen to be really general, 
and not to hold merely for numbers y of the first and second 
class, it is necessary to assure ourselves that Cantor’s first 
and second principles of generation do really * lead to ali 
ordinal numbers (while the third principle defines the 
number-classes). For Schénflies T seemed to state that the 
first two principles only lead to numbers of the first two 
classes, and that we require a new principle to get the 
numbers of the third class. 
However, since this new principle is merely to postulate 
Limes-numbers for an increasing series of ordinal numbers 
also in the case that the series is of type @,, and the concept 
of Limes-number for a well-ordered series t does not pre- 
suppose that the series has necessarily for type a number of 
the second class, it appears that no new principle is required 
to get to any number of the aggregate W of all ordinal 
numbers arranged in order of magnitude. 
This seems to be stated in the ‘Grundlagen,’ pp. 3, 34. 
“ Die Entwickelung....” p. 48 (1900). 
See Math. Ann. Bad. xlix. pp. 218-219; cf. zbid. Bd. xlvi. p. 509. 
*% 
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