ORDER, NUMBER, QUANTITY 259 



terms, whether there is a first or last term, or whether the 

 terms of the series are adjacent. This permits the applica- 

 tion of the notion to infinite as well as to finite series, and 

 to varying types of compactness. The variation in degrees 

 of compactness is very important for science, consequently 

 the postulates as given by Huntington for the definition 

 of these specific types of order will also be given. 



The postulates defining a discrete series are as follows: 



Postulate AT. (Dedekind's postulate.) If Ki and K 2 are any two 

 non-empty parts of K, such that every element of K belongs either to 

 K \ or to K 2 and every element of Ki precedes every element of K 2 , then 

 there is at least one element X in K such that: 



(1) any element that precedes X belongs to K\, and 



(2) any element that follows X belongs to K 2 . 



Postulate 7V2. Every element of K, unless it be the last, has an 

 immediate successor. 



Postulate 7V3. Every element of K, unless it be the first, has an 

 immediate predecessor. 1 



This is the type of order which, if it has a first and a last 

 element, is that exhibited by the gross objects of perception; 

 observed objects are discrete and lumpy and stand adjacent 

 to one another, e.g., the houses on a street, the books on a 

 shelf, and the days in a week. If the series is defined as 

 having a first element but no last element, it is called a 

 progression, and is exemplified by the natural numbers. 



The postulates defining a dense series are as follows: 



Postulate HI. If a and b are elements of the class K, and a < 6, 

 then there is at least one element x in K such that a < x and x < b. 



Postulate H2. The class K is denumerable; that is, the elements 

 of K can be put into one-to-one correspondence with the elements of a 

 progression. 2 



Postulate HI, called by Huntington the postulate of 

 density, forbids one to give empirical examples of this type 

 of series, unless one supposes that the numbers themselves 

 are empirical objects. This postulate requires that there 



1 Ibid., p. 19. 2 Ibid., p. 34. 



