62 Mr. Jourdain on the Transjinite Cardinal 
It is essential for clearness that we should adopt the names 
and notations introduced by Cantor in 1895 and 1897. 
What were before. called “ powers”? were then called (finite 
or transfinite) cardinal nnmbers, the transfinite numbers 
received the fuller designation of transfinite ordinal numbers, 
and finally, the above-mentioned series of cardinal numbers 
received the notation, 
a 8 8S a 
where the suffixes form the whole series of the finite and 
transfinite ordinal numbers. 
It is now easy to state precisely the fundamental results 
of Cantor { with respect to the series (1). 
Ordinal numbers belong only to what Cantor has called 
“well-ordered ” aggregates §, and every part of numbers of 
the whole aggregate of ordinal numbers up to any number, 
arranged in order of magnitude, 
I, 2,..:.V, 5.0,0+1, 042, ...0+¥, 5. 6. 2,052. —ae 
ODL LL OLE hOry Miser aaa 
Gh, OF 1, ys... 2 re 
itself forms a well-ordered aggregate |. The series of 
Alephs (1) is defined as the series of the cardinal numbers of 
well-ordered aggregates, and the Alephs are therefore the 
cardinal numbers of certain of the parts of (2) referred to. 
Cantor has, now, proved, in particular, that &, is the next 
greater cardinal number to Np ; and, in fact, in general9: If 
XN, is any cardinal number of the series (1), then &,41 is the 
next greater; and, inversely: if &, is any number of (1), 
and &, has an immediate predecessor (that is, if y has an 
immediate predecessor in the series (2)), then N, is the next 
greater than this predecessor. If, on the other hand, &, has 
no immediate predecessor (that is, if y is a Limes-number of 
+ “Beitrage zur Begrundung der transfiniten Mengenlehre,” Math. 
Ann, xlvi. pp. 481- 52 (1895), and xlix. pp. 207-246 (1897). See 
especially pp. 481-482, 492, 495, and 497 of the first article, and p. 216 
of the second. 
Cantor had, already in 1883, arrived at many of the formulations 
which were only published much later, (cf. ‘Zur Lehre vom Transfiniten,’ 
Halle a. S., 1890, pp. 11, 12). 
t See ‘Grundlagen,’ ‘pp. 35-39. Cf. Schonflies ‘Die Entwickelung 
der Lehre von den “Punktmanniefaltigkeiten, Leipzig, 1900, pp. 44-50. 
§ ‘Grundlagen,’ pp. 4-5; Math, Ann. xlix. pp 207-208. 
|| CA also Math. Ann. xlix. foot of p- 216, and §§ 2 and 3 below. 
4] A detailed proof of this will be given in the continuation of this 
paper; cf. end of § 10. 
