Miss Wrinch, On the exponentiation of luell-ordered series 233 



The definitions and method used in the earlier part of this paper 

 (**"01 — •341) are suggested in Principia Mathematica *27G. There 

 it is stated tentatively that 



g! p'iT^Ayx . D . ^YiTp^Ayx = min (P«)'X 

 <- g! p\Tp^Ayx . D . PQ'X = prec (P^YX 



The first of these propositions is established in ***1 — •218 : the 

 second seems to be untrue. If in the field of Q there is a term a 

 with no immediate predecessor (as for example the term co if Q 

 were the series of ordinals less than <w + 4), there is a X, a subclass 

 of the field of P***, for which PQ'X is a relation covering with P-terms 

 only the Q-terms which precede a (cp. *"61). In such a case PQ'X 

 is not a P*^ term and so is not prec (P^yx. If P has a last term z, 

 the relation agreeing with PQ'X as far as a and covering a and 

 all subsequent Q places with z will be prec {P'^yx, and therefore 

 the lower limit of X with respect to P^. 



Thus, while agreeing with the proposition if P and Q are well- 

 ordered series and P has a last term, P^ is Dedekindian, and ex- 

 tending it to the proposition if P and Q are well-orde7-ed series, 

 P'^ is Dedekindian tuhen and only when P has a last term, we dis- 

 agree with the conclusion that if P and Q are well-ordered series, 

 P'^ with the addition of a term at the end is Dedekindian even if P 

 has no last term. Instead we would substitute the propositions 

 when P and Q are luell-ordered series, and Nr'Q ^ w, P^ with the 

 addition of a term at the end is DedekiJidian whetJier or not P has 

 a last term, and if Nr'Q •> o), P^ with the addition of a term at 

 tJie end is Dedekindian luhen and only when P has a last term. 



