Miss Wrinch, On the eicponentiation of well-ordered series 219 



On tlie exponentiation of luell-ordered series. By Miss Dorothy 

 Wrinch. (Communicated by Mr G. H. Hardy.) 



[Read 29 October 1918.] 



The problem before us in this paper is the investigation of the 

 necessary and sufficient conditions that P'^ should be Dedekindian 

 or semi-Dedekindian when P and Q are well ordered series. 



The field of P'^ is the class of Cantor's Belegungen and consists 

 of those relations which cover all the members of the field of Q 

 with members of the field of P : several members of the field of Q 

 may be covered with the same member of the field of P, but every 

 member of the field of Q is covered with one member of the field 

 of P and one only. In order to prove that P'^ is Dedekindian it is 

 necessary to prove that every sub-class of the field of P^ has a lower 

 limit or minimum with respect to P^. If there is a last term of 

 the series P'^ it is the lower limit of the null class. Unit sub-classes 

 have their unique members as minima. It remains, then, to con- 

 sider sub-classes with two or more members. 



Now the relation P*'* orders two relations R and *S' by putting R 

 before S, if R covers the first Q-term, which is not covered with the 

 same P-term by both R and h, with a P-term occurring earlier in 

 the P-series than the term with which 8 covers it. Suppose A, is a 

 sub-class of the field of P'^ with at least two members. We will call 

 Qm'^ the first Q-term which is not covered with the same P-term 



by all \'s; and Tp^\ that subset of X which consists of those members 

 of \ which cover Q„/A, with that term, in the class of P-terms with 

 which various X's cover Qm'^; which occurs earliest in the P-order. 



Tp'\ will therefore be contained in \ and not identical with it. It 



will be seen that P'^-terms belonging to Tp'\ come earlier in the 

 P'^-order than terms of \ not belonging to it. Constructing 



Tp'T'X 



we get a smaller subset of \ : members of this subset occur earlier 

 in P''* than other members of X. Continuing this process with 



X, T/X, T/Tp'X, Tp'Tp'iyx,... /*,... V,..., 



we obtain smaller and smaller sub-classes of X: if /a precedes v in 

 this order, members of v occur earlier in the P'^-order than members 

 of fM which are not members of v. We take the common part of 



