frizell: foundations of arithmetic. 397 



66. It is possible to divide R into two ordered 

 classes so that every element of R belongs either to the 

 one or to the other, but neither class has either a first 

 or a last member. For example, the class Ri of all ele- 

 ments whose squares precede 2 and the class Rg of all 

 whose squares follow 2 together exhaust the set R, 

 while neither Ri nor R2 has either a first or a last ele- 

 ment. If we take 4 instead of 2, the element 2 is not 

 included either in R or in Rg ; it separates them. We 

 usually make this separation in practice by selecting a 

 well ordered set, e. g. according to the decimal scale. 

 We take first the highest integer in R^, then the high- 

 est number of tenths, hundredths and so on. Similarly 

 we pick out first the lowest integer in R2 , then the 

 lowest set of tenths, hundredths, etc. 



67. Definition. A well ordered set which has no 

 last element is called a series. 



68. Definition. The series of all symbols of the 

 well ordered set 1 , 2 , . . . w, w -^1, . . . 

 2w, . . . Sw, . . . w^, . . . which pre- 

 cede the element a taken in the above order, is said to 

 be a series of type a . 



69. Definition. An ordered set which has no first 

 nor last element will be called an unbounded set. 



70. Given an unbounded set S ordered according to 

 a rule with respect to which S constitutes an abelian 

 semigroup, suppose that a series K [f ^J of type w has 

 been selected from among the elements of S and let 

 Wi denote the set of all symbols of S which follow, Vg 

 all which precede every fj, let Vi denote the set of all 

 symbols s which precede Wj and Wg the set of all that 

 follow V2 . 



Then one of the sets W^ , V^ must be unbounded and 

 both may be. So also of V2 and W2 . 



3-Univ. Sci. Bull. Vol. V. No. 21. 



