454 Mr. A. E. Harward on the 



for all the transfinite ordinals less than ft, then they are also 

 true for ft. But they have been proved to be true for every 

 ordinal of the 2nd class, therefore they are true for all 

 transfinite ordinals. 



In other words, co? always belongs to the same class as ft 

 when ft is transfinite, and tf y = N y for all finite and trans- 

 finite values of 7. 



It follows from this, of course, that K y = K y (y finite). 



12. The whole of § 9 still holds good mutatis mutandis, if 

 instead of co we take any ordinal number. 



In this way uP is uniquely defined whatever ordinal a 

 and ft may be, and it is proved that a£+ y = a*W. 



It can also be proved, that * 



For if this be true for all ordinals less than y, 

 then, if 7 = 8 + 1, 



= («P)*+ 1 , 



and if 7 have no next predecessor then a Py is the next ordinal 

 after all the ordinals a K (fc<ftry), and (since the law is 

 assumed to be true for all ordinals less than y) it is easily 

 seen that this statement is equivalent to the statement that 

 aPv is the next ordinal after all the ordinals (a' 3 ) 5 (B<y). 

 Therefore a /3y _ r a p\ y? 



and therefore this law is universally true. 



It is easy to show that ©£(.#>©*) belongs to the same class 

 as ft. 



For if this law be true for all ordinals 7(w K <y</3), then, 

 if £ = 8 + 1, 



C(«f)=C(4«0 = C(8)K K 



= C(B) = C(ft); 



and if ft have no next predecessor, then the type of series 

 denoted by wf can be partitioned into a series of type ft of 

 segments each of cardinal number < C(/3). 



.-. CK)<{C(/3)} 2 =C(/3), 



.-. CO£)=C(/9)0S>a, B ); 



and therefore this law is universally true. 



* The order of the symbols must be strictly adhered to. 



