302 SCIENCE PROGRESS 



and the couples are such that in each chain no in or a occurs more than once, and, 

 if a occurs, all ordinal numbers less than a occur also. Thus, a chain is not 

 strictly a part of M, though it obviously determines such a part in a very simple 

 way, and we will say that a chain " exhausts " M if the class of left-hand members 

 (m) of the couples of the chain consists of all the members of M. 



Of course, we do not assume that one of the chains of M exhausts M, or, for 

 example, M lacking some one member : this is what we have to prove. All that 

 is necessary for the validity of what follows is that there are some chains of M, 

 and this is evidently so if M has any members at all, for we can then select 

 arbitrarily chains of M of, say, one and two members. 



A chain carries with it an obvious rule by which it can be well-ordered : this 

 rule is that the couples are to be arranged in the order of magnitude of the right- 

 hand members (a) of the couples of the chain. We shall always assume that a 

 chain is ordered in this way. In conformity with Cantor's language {op. cit. 

 p. 141), a chain P is said to be a " segment " of a chain Q if P is identical with the 

 chain whose members precede some member of Q. In this case, we will also say 

 that Q "continues" or "is a continuation of" P. 



If a class of chains has members of all types less than y, we cannot, in general, 

 conclude that it has one of type y. But if we have a class of chains such that, if 

 x andjy are members of it, and the type of x is greater than that of y, then y is a 

 segment of x ; we can obviously conclude, from a knowledge that the class has 

 members of all types less than y, to the fact that it has a member of type y, pro- 

 vided that y, provided that y has (like the number w) no immediate predecessor. 

 For, in that case, all the members of type less than y build up a chain of type y. 

 We will express the fact that a class of chains is of the nature just considered, but 

 where y need not necessarily lack an immediate predecessor, by saying that it is 

 a class of " direct continuations.'' If, then, (<?i ; a h ai ; «i, «a, «s ; . . .) is a class 

 of direct continuations, the chains being of all types less than «, the class defines 

 (if we take the «th member of the »th chain in the above order) the chain a h 

 « 2 , «s> • • •> and this last chain defines the former class as the class of its 

 segments. 



The whole class of chains of M falls into classes each of which contains all 

 those chains which have the same ordinal type. Now, our first object is to 

 rearrange all these chains in classes of direct continuations such that no chain 

 of M continues the single chain defining and defined by (cf. the end of the last 

 paragraph) any one of these new classes. We then prove that there are ordinal 

 numbers greater than the types of the chains so defined, and yet that each such 

 chain exhausts M. This stage of the proof is necessary if we are to be sure that 

 there is really a chain of M which exhausts M ; and seems to me unavoidably to 

 depend on an argument previously given by me in 1904, which was stated by 

 Russell (1905) — apparently on no grounds save the delusive ones of appearances— 

 to be of a nature quite different from that of Zermelo's principle. 



Firstly, we will see how all the chains of M fall, in a way that is throughout 

 uniquely determined, into classes of direct continuations which we may call 

 " complete," since there is no chain of M which continues the chain defined by 

 any one of these classes. 



Think of all the chains of M of type 1 as forming a class which maybe denoted 

 by U\. With each of these chains of type 1 put, for the moment, all those chains 

 of M which are of type 2, and are also such that the chain of type 1 mentioned is 

 a segment of them. Imagine a chain identical with this chain of type 1 put with 

 each of the chains of type 2 which are thus correlated with it. Though I speak, 



