On a type of completeness characterizing the general laws for separation of point-pairs
Cooper Harold Langford · Transactions of the American Mathematical Society · 1927
In a forthcoming paper by E. V. Huntingtont a number of sets of postulates or determining conditions for the type of order called of have been given. These sets are selected from a list of general properties which characterize reversible order on a closed line, and each of the sets is shown to imply all the others so that the several selections are equivalent. It is to be shown in the present paper that sets of postulates for separation of point-pairs are characterized by a property which is closely analogous to ordinary completeness. A class of propositional functions will be defined, to be called general laws,tto which any member of a set of postulates for this type of order belongs, and it will be shown that such sets are sufficient to determine the truth or falsity of any general law which can be constructed on the base K, R4, the base for the set. This is a question of deducibility; one or the other of every pair of mutually contradictory general laws on K, R4 must be deducible. The question of deducibility arises here in the following manner. It seems to be true from inductive considerations that each of these sets is a sufficient characterization of the type of order in question and thus that the theorems which follow from any one of them might be held to be exhaustive of the general properties which are understood to attach to systems involving separation of point-pairs. Any such set might then be taken as a set of defining properties for separation of point-pairs in the sense that any theorem which is commonly understood to hold for this type of order is implied by the postulates and no theorem which is recognized as not belonging to this type of order does follow from the postulates. In this sense