64 Mr. Jourdain on the Transjinite Cardinal 
for all finite ordinal numbers vy, then 
[2N.: 
and so on for all the numbers of (1). From this reasoning, 
which first appeared in a published form in a paper in which 
Hardy + constructed an aggregate of points of cardinal 
number S; in the continuum, it follows that every cardinal 
number is either contained in the series of Alephs (1), or is 
greater than any Aleph. 
If, now, Cantor’s { view that every cardinal number is an 
Aleph —which he expressed in the equivalent form that every 
well-defined aggregate can be put, by re-arrangement if 
necessary, in the form of a well-ordered ager egate—is to be 
substantiated, we must prove that the supposition that a 
cardinal number is greater than any Aleph leads to a con- 
tradiction. 
2 
sal @ 
If a cardinal number were greater than any Aleph, it 
would be equal to or greater than the cardinal number of 
the series (1) of all Alephs. For to this series can be cor- 
related § the series (2) of all ordinal numbers, and every 
Aleph is the cardinal number of some segment || of the 
series (2) ; the cardinal number in question must, then, be 
at least equal to the cardinal number of the whole series (2), 
and, consequently, to that of the whole series (1). 
If, now, the series (2) and consequently (1), be well- 
ordered, the ordinal type of (2) (or (1)) is an ordinal number, 
8, and the cardinal number of (1) (or (2)) is an Aleph, Ne. 
Further, this ordinal number 8 must be the greatest ordinal 
number, and, consequently, Ng must be the greatest Aleph. 
But there can be neither a greatest ordinal nor a greatest 
Aleph; for, given 8, the type of the aggregate (1...8) is 
the ordinal neanioet 8+1, and 
Beas. 
Neii > Ne: 
This contradiction was first published by Burali-Forti 4, 
“A Theorem concerning the Infinite Cardinal Numbers,” Quart. 
Journ. of Math. 1903, pp. 87-94. 
t ‘Grundlagen,’ p. 6. 
§ This word is to imply that the two aggregates are similarly ordered 
in the sense of Math. Ann. xlvi. p. 497. 
| I use this word to translate Cantor’s ‘ Abschnitt’ of Math. Ann. 
mix pp) 20, 
© “Una questione sui numeri transfiniti,” Rend. del circolo mat. di 
Palermo, xi. (1897). 

