Numbers of the Exponential Form. 53 



ordinally similar to the series o£ Alephs, or. what is the same 

 rhino-, to the series of class-characteristics : 



Hence, to every class-characteristic co y of Cantor's ordinal 

 numbers corresponds one. and only one. Cantor's ordinal 

 number y, and vice versd. Thus, if' 



/3=ft) a , 



* cannot be a Cantor's ordinal number, for, it' it were, /3 would 

 be one too. Further, (3 or w a (if it exists) is the least ordinal 

 number which is greater than all Cantor's ordinal numbers, 



Accordingly, if /3 exists, the third principle is satisfied, in 

 spite of first appearances, by the series of all Cantor's ordinal 

 numbers; and the (Burali-Forti's) contradiction resulting 

 herefrom leads us to deny the existence of /5, the type of W. 



Xow, the series W is well-ordered*, although it cannot 

 have a type, and evidently other well-ordered series (having 

 do types) transcending W, can be formed. So we must 

 conclude that the series W is similar to a segment merely of 

 the series (2J[{) *uch that every well-ordered series is similar 

 either to it or to a segment of it |- 



We can define a series ordinally similar to W by positing 

 one element and then positing successive elements according 

 to Cantor's first and second principles. It results from our 

 considerations that the ordinal number of every element 

 thus formed is subject to Cantor's third principle ; that is to 

 say, we cannot, without contradiction, speak of an ordinal 

 number of an element which follows all those whose ordinal 

 numbers obey the third principle. In other words, we cannot, 

 us seemed possible if we assumed that 



co y >y 



always, define ordinal numbers which transcend all Cantor's 

 ordinal numbers. The name of "principle of limitation" 

 may, then, convey the wrong impression that the series W is 

 not, as we shall say in the nexi section, "absolutely" 



infinite ;. 



* Phil. Mag. Jan. 1004, pp. (i.j_Gb\ 



t This is the series described, in not quite such aecurate terms as the 

 above, in Phil. Mag. Jan. 1904, p. 67, lines 18-19. It follows from 

 the above that 03 can not be used as a substitute for W in a criterion 

 of •" consistency." 



t The "absolute" infinity (if W was stated by Cantor in 1882 

 (• Grundlagen . . ..' p. 14;. 



