Certain finite nonprojective geometries without the axiom of parallels

Abstract

In this paper, and upon, are undefined terms. The phrases, is on a is incident upon a is incident upon a point, is on a point, are all to be considered synonymous. We will say that a line I intersects a line k (at a point P) if and only if P is on both I and k. Two lines are parallel if and only if there is no point which is on both. The following axioms will be used: AXIOM I. If P and Q are distinct points, there is exactly one line on P and on Q. AXIOM II. If I is a line, there is at least one point not on 1. AXIOM III. If I is a line and P is a point not on 1, there are exactly m distinct lines on P (m ?2) which are parallel to 1. AXIOM IV. There is at least one line with exactly n points on it, n _2. The entire set of points and lines whose existence is postulated by these axioms (for given m and n) will be called a geometry. Other work done with an axiom system containing the Axiom III as here stated is not known to this investigator, but Szamkolowicz [3 ] has reported on an equivalent system, and similar systems have been studied [2], [4]. As they are here stated, the axioms may or may not be consistent, depending on the values assigned to m and n. For n =2, their consistency for any m is established by the existence of a model constructed by A. N. Milgram [4], and for n = 3, m = 4, their consistency is shown by a model described by Abraham Barshop [I ]. This paper will demonstrate their incon sistency for m=2 and n>2.