408 KANSAS UNIVERSITY SCIENCE BULLETIN. 



ural numbers in § 94 are only a part of these series. 

 But they are also only a part of the permutations of 

 the whole set of natural numbers. Other permutations 

 are obtained by the following Lemma. From any tv 

 series of symbols may be obtained a set of permuta- 

 tions of the symbols forming a series of type tv' = iv"\ 



Proof. Letai, ag, . . . aN, • . . denote the given 

 symbols in the given order. Without changing the 

 order of the higher a's put ai successively in every sub- 

 sequent place. This set of permutations is obviously of 

 type w . 



From every one of them by repeating the process on 

 a 2 we obtain again a w;-series .'. in all a series of per- 

 mutations of type td Repeating the process with as, 

 a4, . . . . successively we have a series of permuta- 

 tions whose type is w'. Q. E. D. 



97. Thus the arrangements of the natural numbers 

 furnish a series of type not lower than W. For by § 96 

 we first deduce from the series 1, 2, 3, ... . i. e. , the 

 normal order, a set of permutations forming a series of 

 type w'. That is, to every ordinal symbol preceding iv' 

 is assigned a permutation, and vice versa. Then by >i 

 94 arrange the natural numbers in series iv' which ob- 

 viously is a different permutation from any of the pre- 

 ceding •.' it can not be obtained by the process of ^ 96. 

 Now repeating the method of ^:J 96 on each ?('-series of 

 this permutation we use up all ordinal symbols between 

 w' and that which results from it by a second applica- 

 tion of § 94. This holds step by step as long as the 

 latter process can be carried on. Thus to every ordinal 

 symbol in succession preceding W is assigned a new 

 permutation. That is, we have a set of permutations 

 of all the natural numbers forming a series whose ordi- 

 nal type is W. 



98. The process just described for making permu- 

 tations of all the natural numbers can not yield a series 



