450 Mr. A. E. Harward on the 



It also follows from the definition that if this proposition is 

 true when f3 = y, it is also true when /3 = <y + v (v finite). 



I proceed to prove that if the proposition is true for every 

 ordinal less than /3, it is also true for /3 in the case where /3 

 denotes a type of series with no last term. 



In this case a> a+/3 denotes the first ordinal after all the 

 ordinals col (y<a + /3). 



The series of ordinals preceding <w a+/3 (which is a series of 

 the type denoted by co a+l3 ) can be divided as follows into an 

 aggregate of segments each of type ft> a , arranged in a series 

 of type &>£. 



(1) 1,2,3 7l (7i<«> a ). 



(2) <» a •••• y 2 (72<*> a 2). 



(3) co a 2 7 3 (73<*> a 3). 



(8) co a 8(S<cot) 7 { r <a) a (S + l)[ 



In this series 8 will among other values take all the values 

 ft) v (y</3), and since co a coy = co a+ y if y i s less than y8 this series 

 includes all the ordinals co y (y<ot + /3), and since the type of 

 series denoted by /3 has no last term, it is clear that each 

 term of this series is less than some ordinal o>T(7<a + /3). 

 Therefore this series is the series of ordinals less than co a+ P. 



.'. CO a+f * = CQ a CD( 3 . 



Therefore this proposition is true for all values of /5. 



10. I next prove that if /3 be any ordinal of the 2nd class, 

 then co& is also of the 2nd class. 



If &/ belongs to the 2nd class it is obvious that a> y+1 also 

 belongs to the 2nd class, for 



C(a>v+i) = C(a) y a)) = G(a)0^o 



If ft (of the 2nd class) denote a type of series with no last 

 term, and if the proposition that C (co y ) = C(y) is true for all 

 transfinite values of 7 less than ft ; then it can be shown 

 that this proposition must be true for ft also. 



For the series of ordinals less than co? is partitioned by the 

 ordinals 



1, co, ft/, co\ ...to", ft> w+1 , ...a/, ... (y<ft) 



into segments of which these ordinals are the first terms. 

 Each segment is a series of type ft/ (y<ft), and therefore 



