The parallel postulate

Abstract

This is a survey of what is known regarding weaker versions of the Euclidean parallel postulate, culminating with a splitting of the parallel postulate into two weaker and independent incidence-geometric axioms. Among the weaker versions are: the rectangle axiom, stating that there exists a rectangle; the Lotschnittaxiom, stating that the perpendiculars to the sides of a right angle intersect, and Aristotle’s axiom, stating that the distances between the sides of an angle grow indefinitely. Several statements that are equivalent, with plane absolute geometry as a background, to each of these axioms, as well as an analysis of their syntactic simplicity are presented. The parallel postulate is found to be equivalent to the conjunction of the following two axioms: “Given three parallel lines, there is a line that intersects all three of them" and “Given a line a and a point P on a, as well as two intersecting lines m and n, both parallel to a, there exists a line g through P which intersects m but not n."