frizell: foundations of arithmetic. 407 



the elements of any ii;-series may be rearranged so as 

 to produce a series of type w "' . Applying this method 

 successively to the ^(;-series in the preceding we obtain 

 in place of ?<; , 2w, . . . w', 2w' . . . that is, instead 

 of 2(;^ wegeta type?^i(;'. Thenw^^iv, iv^-\-2tv, . . . 

 are replaced by !(; it'' + ^(;', ivw' -{-2w' , . . . making 

 in all type 2ivw' replacing the former 2w^. 



Then come in succession types 3ww' , iww' , . . . 

 so that the original iv^ expands into a type w^w' . It 

 is clear that in this way we retrace precisely the steps 

 of the process of § 22, building up on each new symbol 

 the abelian semigroups according to the lower and low- 

 est rules. 



Therefore by successive applications of the method 

 it will be possible to rearrange the natural numbers in 

 series of types as high as any of the set hitherto de- 

 fined. This result may also be stated. The transfinite 

 ordinal series so far defined may each be put into one 

 to one correspondence with the set of natural numbers. 



95. The symbols w, tv-\-l, tv-\-2, . . . i. e., the 

 transfinite symbols, are said to form the second ordinal 

 class, the finite symbols constituting the first class. 

 The latter class was said to be of type w, where w is 

 the symbol following next after all the finite symbols 

 of K[^to]. Likewise we shall say that the first and 

 second ordinal classes together form a series of type 

 W , introducing this new symbol without assigning to 

 it any properties except that it shall follow next after 

 all symbols of the second ordinal class, i. e., after all 

 the setK['M;''] : w, w', w" , . . . 



96. We see that the ordinal symbols play a double 

 role. They were defined as symbols forming a well 

 ordered set. But to each symbol which has no imme- 

 diate predecessor corresponds a series of which it is the 

 type, viz.: the series of all its predecessors or any 

 ordinally similar series. The arrangements of the nat- 



