52 Mr. Jourdain on Transfinite Cardinal 



merely of 223- The ground of this lies in the fact that 

 Cantor's third, or limiting principle, which applies to all 

 ordinal numbers, does not apply to certain well-ordered 

 aggregates, which transcend even the series of all the transfinite 

 ordinal numbers of Cantor. 



In order to state shortly what is contained in the third 

 principle, it is convenient to single out the first number of 

 each of the number-classes as the " class-characteristic " of all 

 the other numbers of that class We thus define the "class- 

 characteristic " of any ordinal number a as either a itself, if 

 a is the first number of a number-class (a = w y ) *, or, if not, 

 the first number (® y ) after a which is the first of a number- 

 class. 



Then the principle in question can be stated : — 



The cardinal number of all the ordinal numbers preceding 

 the class-characteristic co y of a given ordinal number is tf y . 



Let us now consider whether the series of all the ordinal 

 numbers which are subject to the third principle has a type : 

 in other words, whether the assumption that it has a type 

 leads to a contradiction, as was the case in Burali-Forti's 

 argument. Let the type be ft : then ft is its own class- 

 characteristic t, say /3 = <y a . To find the cardinal number of 

 all the ordinal numbers preceding ft, we notice that every 

 Aleph less than X^ (that is to say, every Aleph whose suffix 

 is less than ft) is the cardinal number of some segment of the 

 series of type ft, so that the cardinal number in question is 

 at least equal to Kg. That it is also at most equal to K^ is 

 evident from the fact that N^ is the next greater Aleph to 

 the series of Alephs of all the segments. Thus the cardinal 

 number of the ft ordinal numbers is 



Up ov **av 



and, since it is not tta, the third principle does not appear to 

 be satisfied. 



However J, although co a can never be equal to « when a is 

 a Cantor's ordinal number, it does not follow that ft is not 

 equal to osp. And. in fact, this is so, as the following 

 considerations show. 



The series of Cantor's ordinal numbers is known to be 



* See the notation in Phil. Mag. March 1904, p. 295. 



f For if /3 is not the first number of a class, there are predecessors of 

 the same class. But every predecessor of /3 belongs to one of Cantor's 

 number-classes which is itself surpassed by a Cantor's number-class. 



| My attention was called to this point, which I had overlooked, by a 

 remark of -Mr. G. H. Hardy, Fellow of Trinity College, Cambridge. 



