frizell: foundations of arithmetic. 409 



of type higher than W since, as we have seen, it gen- 

 erates precisely the series W of ordinal symbols. That 

 there are types higher than W is obvious, for we can 

 proceed with W just as with tv to generate new e-sets 

 and new abelian semigroups, and there is no limit to 

 the possibilities in the way of still higher symbols and 

 rules. Now it is quite conceivable that there may be 

 further permutations of natural numbers not obtainable 

 by the above process. As a first step toward investi- 

 gating this question let us consider the simpler one 

 whether the natural numbers themselves can be ar- 

 ranged in a series of ordinal type higher than W. For 

 this purpose we will establish the following 



Lemma. Permutations of the natural numbers can 

 be made by the process described and ordered so as to 

 form a series of type higher than any series of all the 

 natural numbers, however arranged. For by the proc- 

 ess in question we can always form a ne w permutation 

 which differs from the first permutation in at least its 

 first element, from the second in at least its second 



element from the ( i^ + 1 ) st in at least its 



( w; + 1 ) st element, and so on, therefore is not included 

 in any set of permutations ordinally similar to any 

 possible arrangement of all the natural numbers. 

 From this Lemma we readily obtain the 



Theorem. Every possible arrangement of the natural 

 numbers is a series of the second class. For by the 

 Lemma to every such arrangement in series can be as- 

 signed a set of permutations forming a series of higher 

 ordinal type. But by § 97 the process by which this is 

 eflFected yields a set of permutations forming a series of 

 type W. Therefore every possible arrangement of nat- 

 ural numbers in series is of type lower than W, there- 

 fore belongs to the second ordinal class ( being ipso facto 

 >w). q. E. D. 



