448 Mr. A. E. Harward on the 



aggregates o£ ordinals, comprising all ordinals, and the 

 ordinals would be an aggregate. 



The class of classes o£ ordinals is therefore an unlimited 

 well-ordered serial class similar to the class of ordinals. 



The same is true of the class of associated cardinals. 



Every transfinite cardinal is included in this class of 

 cardinals because every aggregate can be arranged in a well- 

 ordered series. 



The first ordinals of the classes of transfinite ordinals are 

 denoted by the symbols 



0), ft> : , G> 2 , . . . ftj w , ©„, + !, . . . CO/3, . . . 



(/3 any transfinite ordinal). 



The class of which co is the first is the 2nd class, the class 

 of which coi is the first is the 3rd class, the class of which w v 

 is the first (y finite) is the (v + 2)th class, the class of which 

 &) w is the first is the &>th class, and the class of which «„ is 

 the first (/3 transfinite) is the (Bih. class. 



The cardinal associated with the 2nd class (of which co is 

 the first ordinal) is ft and the cardinal associated with the 

 class of which co^ is the first (/3 finite or transfinite) is 

 designated by the symbol ft/3. 



It is implied by what I have proved above that there can 

 be no cardinal intermediate in value between ftp and ft/s+i. 

 But as the point is of great importance it is as well to restate 

 the proof. 



The elements of an aggregate of cardinal number ft# can, 

 by definition, be arranged in a series of type co^, and the 

 elements of an aggregate of cardinal number ft^+i can be 

 arranged in a series of type top+i. 



Let the elements of any other aggregate be arranged in a 

 well-ordered series, call the type of that series <y. 



Then, if y < w 



C(7) ^fft/3 (it is in fact < , but it is not necessary to assert 

 this) , 

 if y>G>0+„ 



0(7)>fc+i, 

 if » /B <7<^ +1 » 



then 7 belongs to the same class as top and C(y) = ftp. 



It is easy to show that the cardinal number of the aggregate 

 of ordinals belonging to any class is equal to the cardinal 

 number associated with the next following class. For let x be 

 the cardinal number of the aggregate of ordinals belonging 



