Transfinite Numbers. 447 



the symbol C(/3) to denote the cardinal number of the 

 aggregate of terms of a series of type (3 (/5 = any ordinal). 

 It is obvious that if /3< 7, C(/3) <C(y), because the aggregate 

 of terms in a series of type /3 is similar to a part of the 

 aggregate of terms in a series of type 7, and therefore (by the 

 definition of u less than " for cardinals) the cardinal number 

 of the latter ao-oreoate is not less than the cardinal number 

 of the former. Therefore it is equal or greater. 



The first class of ordinals consists of the finite ordinals. 



Two transfinite ordinals /3 and 7 are said to be of the same 

 class if C(/3) = C(y), and the cardinal number C(/3)( = C(7)) 

 is said to be the cardinal associated with that class. 



If j3 and 7 are of the same class then all the intermediate 

 ordinals belong to that class. 



Forif/3<a<7 



then C(/3)<C(S)<C( 7 ) ; 

 therefore if C{/3) = C(r/h 



then C(S)=C(/3) = C(7). 



The classes of the ordinals are therefore segments of the 

 serial class of ordinals, and therefore form a simply-ordered 

 serial class, and this class of segments - must be well-ordered, 

 for if there were any aggregate of segments with no first 

 segment it would be possible by taking one ordinal from each 

 to obtain a progression of ordinals in descending order of 

 magnitude. 



The cardinal associated with any class is greater that any 

 cardinal associated with any preceding class, for it is by 

 definition not equal to any of those cardinals, and it cannot 

 be less because if fi>y, C(/3)^C(y). 



Therefore the cardinals associated with the classes of trans- 

 finite ordinals form a well-ordered serial class similar to the 

 class of classes of ordinals. 



There cannot be any last class of ordinals, because if there 

 were the associated cardinal would be the greatest cardinal*. 



Therefore the ordinals belonging to any class form an 

 aggregate, for they are part of the aggregate of ordinals pre- 

 ceding the first ordinal of the next class. 



The classes of ordinals are not a series (i. e. not an aggre- 

 gate), for if they were so, there would be an aggregate of 



* It should "be observed that it is at this stage of the argument that I 

 for the first time make use of the axiom assumed in § 3, for the proof 

 that there is no greatest cardinal depends on that axiom. 



