Sets of independent postulates for betweenness

The " universe of discourse " of the present paper is the class of all welldefined systems (K, R) where K is any class of elements A, B, C, ..., and R is any triadic relation. The notation R [ABC], or simply ABC, indicates that three given elements A, B, C, in the order stated, satisfy the relation R. Examples of such systems (K, R) are the following, of which example (a) is the most important: (a) K is the class of points on a line; AXB means that the point X lies between the points A and B. (b) K is the class of natural numbers; AXB means that the number X is the product of the numbers A and B. (c) K is the class of human beings; AXB means that X is a descendant of A and an ancestor of B. (d) K is the class of points on the circumference of a circle; AXB means that the arc A-X-B is less than 180O. (e) K is a class comprising four elements, namely, the numbers 2, 6, 6, an 6 A X Ce A 1V1) 74= A .T) anu O?o; LLa n mieans~ A =1 At D. It is obvious that these systems, and others like them, will possess a great variety of properties expressible in terms of the fundamental variables K and R. The object of this paper is to state clearly the characteristic properties of the type of system represented by example (a) above, by which this type of svstem is distinguished from all other possible systems (K, RI). In Section 1, we give a basic list of twelve postulates, due essentially to Pasch,t from which various sets of independent postulates will later be selected.