Miss Wrinck, On the exponentiation of ivell-ordered series 221 



the ivhole of the Q-terins with P-terras. When PQ'X does nut 

 cover the whole of the Q-tenns, but covers Q-terms only up to 

 z (say), all p's will agree in their covering of Q-spaces up to z, and 

 the remaining Q-spaces will be covered differently by different 

 members of p. To get a maximum of the p's with respect to P*^, 

 we want a relation S which is a p such that no member of p comes 

 later in the P^-order. Now. if P has no last term, every P-term 

 is followed by other P-terms. However S covers z and the Q-spaces 

 after z, by replacing the term covering any member of the field of 

 Q after ^ by a member of the field of P following it in the P-order, 

 we obtain a relation T which is a p and follows ;Si in the P'^-order. 

 ;S' is, consequently, not the maximum of p in the P^-order. Now 

 if z in the field of Q is covered by PQ'X, the term innnediately 

 following z will also be covered by PQ'X. Therefore, if Q is a finite 

 series or an co, PQ'X will always cover the whole of the Q-terms ; 

 since, as X has at least two members, it will always cover one Q- 

 term. Any X will then have a lower limit or minimum with 

 respect to P*^. In such cases, P*? will certainly be Dedekindian 

 with the addition of a last term, whether P has a last term itself 

 or not. 



But if Nr'Q is greater than o), it is possible to find a subclass 

 X of the field of P^ which is such that PQ'X does not cover the 

 whole of the field of Q. 



For, let 1 and 2 represent the first and second terms in the P- 

 series and let (e.g.) 



i...hi(r)2...h2(aiii 



represent a relation which covers the first ^ Q-terms with 1, sub- 

 sequent terms up to (but not including) the ^th term with 2, and 

 all remaining terms with 1. Such a relation is clearly a member 

 of the field of P'^. Consider the class of I'elations X which cover 

 all Q-spaces up to z with 1, and all the Q-spaces following z with 

 2, as z is varied from the second Q-term to the ^th, where ^ is an 

 ordinal number with no immediate predecessor. We will arrange 

 this class of relations in the P^-order. 



1... 1-1(^)2. ..h2(0,22.... {^<0 



11112 h2(f), 22 



11122 f-2(0> 22 



11222 ^-2(0> 22 



12222 ^-2(0, 22 



This class has no minimum in the P'^-'-order, and PQ'X covers all 



