frizell: foundations of arithmetic. 405 



rule has not been made a rule of combination for any 

 whole set. This would involve assigning properties to 

 it and completing the arithmetic of the symbols we 

 have defined, which is beyond the present purpose. It 

 is enough to have them in series ; that forms the foun- 

 dation of their arithmetic. • 



92. The mere existence of the series of symbols de- 

 veloped in the preceding § § is enough to solve a great 

 variety of problems arising in analysis, the nature of 

 which will be illustrated by comparing this series with 

 series consisting of absolute numbers. The natural 

 numbers form a series of type w if they are arranged 

 in the order of their genesis. It is, however, possible 

 to arrange them in series of higher types for e, g. the 

 odd numbers alone form a series of type w. If to this 

 we add on the even numbers successively we obtain 

 series of types 



w-\-l, w-\-2, . . . ii^ + N, . . . 

 and thus the whole set is ordered in type 2w. 



If we order the set R of § 60 as follows : 



(1,1), (2,1), (3,\). (4,1), . . . 

 (1,2), (3,2), (5,2), (7,2), . . . 

 (1,3), (2,3), (5,3), (7,3), . . . 

 we have a series which can be put ordinally into 

 one to one correspondence with the series 1,2, . . . 

 w-^1, w-\-2, , . . 2w, 2w-\-l, . . . i.e., 

 is of type w^ . 



The same set of numbers (the common fractions) 

 can, however, be arranged in a series of higher type. 

 For every finite, simple, continued fraction is equal to 

 some common fraction and conversely. Now let us 

 order the finite continued fractions according to the 

 values of their successive quotients 



Thus the continued fractions containing each a single 

 quotient form a series of type w . Then from each 



