ESSAYS 303 



for the sake of what seems to me ease of visualisation, ot many chains identical 

 with a certain chain, it must not be forgotten that there is really only one chain 

 which is identical with a given chain — namely that chain itself. 



We do this for all the chains of M of types 1 and 2, and thus get a class « 2 such 

 that, if y is a member of u 2 , y is a class of direct continuations, and the continua- 

 tions are certain chains of types 1 and 2. It is to be noticed that there is, 

 throughout this construction, no choice of particular members of the u's ; and each 

 u except the first is formed from the preceding u or u's and the chains of M in a 

 completely non-arbitrary and uniform way. Also notice particularly that the u's 

 are such that the members of u n contain those of u n -i in such a way that those of 

 u n grow out of those of z/„_i by addition to all the latter ones, on a perfectly 

 definite system, of all the chains of type n. We can thus see that the u's men- 

 tioned mark stages in one and the same process of building up, out of chains of M, 

 classes of direct continuations ; each u n , namely, contains all possible classes of 

 direct continuations where the chains are of all types from 1 to n. 



Suppose that those chains of M whose types are all the ordinal numbers less 

 than y, where y has an immediate predecessor y— 1, thus form classes of direct 

 continuations ; the class u y -\ being such that, if x is a member of u y -i, x is a class 

 of direct continuations where the continuations are of types 1, 2, 3, . . . , y- 1. Of 

 the chains of M of type y — if there are any — put in each x all those which con- 

 tinue the chains in that x. Then imagine that chains identical with all those 

 previously in x are put with the appropriate chain of M of type y, so that a new 

 class of direct continuations of types 1, 2, 3, . . . , y arises ; the class of these 

 classes may be denoted by u y . Here again, this u y is determined uniquely by the 

 preceding u's and the chains of M of type y — if, indeed, there are any. Also we 

 imagine several sets of chains identical with a given set, just as above we 

 imagined several chains identical with a given chain. 



Suppose that no u of suffix less than a> is null, and consider all those classes 

 of direct continuations formed by the above definite process which are such that 

 all u's of suffixes less than o> appear in the process. We can consider the whole 

 class of the classes thus formed by this infinite process, because it is a complete 

 and well-defined mathematical object. In general, if y has no immediate pre- 

 decessor, we can evidently form a chain of type y out of a class of direct continua- 

 tions where the continuations are of all types less than y. It must again be 

 emphasised that we are able to conclude here from chains of types less than y to 

 a chain of type y, without a use of Zermelo's principle, only because all these 

 chains are direct continuations. Thus, if y has no immediate predecessor, and if 

 no u of suffix less than y is null, u\y is not null. 



This unique determination of a whole transfinite series of u's, continuing as 

 long as there are any chains of M left, is the most important part of this construc- 

 tion. It will be seen that the method of generation of the series of u's corre- 

 sponds to the two " principles of generation " spoken of by Cantor {op. cit. pp. 

 56-60, 169) ; the combined action of these two principles carries us to -all the 

 number-classes of Cantor in succession. 



Secondly, any one of these complete classes of direct continuations determines, 

 and is determined by, a single chain of M. 



Thirdly, we will piove below that, for this chain, there is at least one ordina. 

 number £ such that no segment of the chain or the chain itself is of type £ ; 

 and yet the chain exhausts M. This chain is, then, obtained without a use of 

 Zermelo's principle. Indeed, the class of all chains of M, where we do not 

 assume that any of its members exhausts M, falls, without any arbitrary choice 



