304 SCIENCE PROGRESS 



on our part, into a class of chains, each of which can be proved to exhaust M for 

 some definite ordinal type. 



Consider any one of the complete classes of direct continuations. The chain 

 defined by it must exhaust M ; for otherwise there would be a member of M 

 which was not a member of the chain, and in that case the class of direct continua- 

 tions in question would not be complete. Hence we see that the chain just 

 referred to cannot be a segment of any other chain of M, and, now that we know 

 that such chains exist, we can say that the chains which exhaust M make up the 

 remainder of the class of chains of M which is obtained when we cancel all those 

 chains which are segments of other chains. 



But there still remains an apparent possibility : may it not happen that, how- 

 ever great the ordinal number | may be, a complete chain is always such that it 

 has a segment of type | ? But this is impossible, for a chain of such a nature that, 

 however great £ may be, it always has a segment of type £, must be of the type 

 (8) of " the series of all ordinal numbers." Now, we can prove that this series is 

 well-ordered, for any part (P) of it which has any terms at all—say p — has a first 

 term — namely, the first term of the well-ordered series formed by p and those 

 terms of P which precede/. Hence 8 is an ordinal number, and hence 3 is both 

 the ordinal number of a series and a term of the series, so that 8 is greater than 8. 

 This implies, of course, that the series of all ordinal numbers is ordinally similar 

 to a segment of itself, and thus that the series is not well-ordered ; and, therefore, 

 that there is no such thing as what we meant to denote by the phrase " the series 

 of all ordinal numbers," which would thus be both well-ordered and not well- 

 ordered. But at present we only need the proof that it is impossible that a com- 

 plete chain should have segments of all types. This is a form of the argument 

 on which I laid stress in 1904. 



It must be noted that the proof given in the last paragraph essentially depends 

 on the fact that a class of direct continuations has been obtained. It is only for a 

 class of direct continuations, unless we use Zermelo's principle, that we can con- 

 clude that, if, for any £, a member of the class is of type £, then there would be a 

 member of the type of " the series of all ordinal numbers " ; just as, without a use 

 of Zermelo's principle, we cannot conclude, from the fact that a chain of chains has 

 members of all types which can be defined by " mathematical induction," and 

 which we usually call " finite," that it has a member of type a>, unless it is a class 

 of direct continuations. 



We have, then, proved that, for any M there is at least one chain that both 

 exhausts M and is of type less than some type £, say. Of the ordinal numbers 

 greater than the type of this chain there is one that is the least, since ordinal 

 numbers in order of magnitude form well-ordered series ; let it be denoted by C 

 This ordinal number £ must have an immediate predecessor ; for, if it had not, the 

 chain itself would be of type f, and so £ would not be the least ordinal number that 

 is greater than the type of the chain. Hence f is of the form P+i. Now, f 1 is 

 the first number of one of Cantor's number-classes ; for if there were numbers of 

 the same number-class which were less than the ordinal number just mentioned, 

 this ordinal number would not be the least to which would belong chains which 

 were not continued by other chains of M. 



So any aggregate M can be well-ordered, and thus Zermelo's principle can be 

 proved. The problem may be considered as one of organisation, and it is rather 

 remarkable that it should be possible so to organise the members of any infinite 

 class that selections of series of any types less than a maximum one can always 

 be made from it. This is very useful for many mathematical purposes. 



