Non-Desarguesian planes and weak associativity

Abstract

During the 19th century various criticisms of Euclid's geometry emerged and alternative axiom systems were constructed. That of David Hilbert ([1], 1899) paid particular attention to the independence of the axioms, and it is his insights which have shaped many of the further developments during the 20th century.We can, from his insights, define an affine plane as a set of points, with distinguished subsets called lines such thatAxiom 1: Given two distinct points, there is a unique line containing them both.Axiom 2: Given a line L and a point, p, not contained in L, there is a unique line containing p which does not intersect L.Axiom 3: There exist at least three points, not belonging to the same line.