222 Miss Wrinch, On the exponentiation of well-ordered series 



the Q-places up to the ^th with 1 and does not cover the subse- 

 quent Q-places at all. It is therefore not a member of the field 

 of P^. But, as we have seen, every relation which agrees with 

 PQ^X as far as it goes, and covers the other Q-places with any P- 

 terms whatever, precedes all X's : and any member of the field of 

 P^ following this relation, and all relations agreeing with PQ'X as 

 far as it goes, follows at least one member of X. Thus, e.g., the 

 relation 



ll...hl(^), 2111 



precedes all X's, and any relation following it and all relations 

 agreeing Avith PQ'X as far as it goes (as e.g. the relation 



11211... h 1(0, 21211...) 



follows at least one relation belonging to X, e.g. the relation 



11122... \-2{0, 222... 



Thus \ will have a lower limit if and only if there is a maximum 

 among the relations covering all places up to the ^th with 1. 

 And this is the case when and only when P has a last term u (say). 

 For then the relation 



111 \-l{^)uiiu... 



will be the lower limit of X. Thus if Nr'Q is greater than o), it 

 will be the case that all existent sub-classes of the field of P'^ will 

 have a lower limit or minimum when and only when P has a last 

 term. A non-existent subclass (i.e. a subclass with no members) 

 will have a lower limit or minimum when and only when P has a 

 last term. If Nr'Q is greater than co, P^ is Dedekindian when P 

 has a last term, and if P has no last term P^ even with the addition 

 of a last term is not Dedekindian. We thus arrive at the following 

 conclusions. When P and Q are well-ordered series, (1) P^ is 

 Dedekindian when and only when P has a last term ; (2) if Nr'Q 

 is greater than co, P*^ with the addition of a last term is Dede- 

 kindian if and only if P has a last term ; (3) if P^ is made Dede- 

 kindian by the addition of a last term when and only when P has 

 a last term, Nr'Q is greater than co. 



These propositions will now be established. 



[The symbols used are those o/Principia Mathematica. Among 

 the propositions referred to, those whose nwnhers are greater than 

 1 are proved in P.M., ivhile the others are established in the course 

 of this paper.'] 



*01. QjX = mmQ'y{s'X'y^eQKjl) Df 



*-02. Tp'X = XnM {M'QJX = mmp's'X'Q^,'X) Df 



