A theorem on parallels

Abstract

For the theorem which we wish to prove we need Hilbert's axioms of incidence I 1-3 referring to plane geometry and the axioms of order [2, pp. 3-5]. Special emphasis will be placed on the second axiom of order, II 2: If A and C are two points of a straight line, there exists at least one point B of the line so situated that C lies between A and B. Now the main purpose of our theorem is to show the existence of parallels using continuity and not congruence as is usually the case in plane geometry. Therefore we have to state our axioms of continuity in a form not involving congruence, known as the postulate of Dedekind [1, p. 23 ]: For every partition of all the points of a segment into two nonvacuous sets, such that no point of either lies between two points of the other, there is a point of one set which lies between every other point of that set and every point of the other set. We prove the following