Sharpening independence results for Huntington's affine geometry

Abstract

We improve upon Huntington's affine geometry by showing that his independence proofs can be, in some cases, simplified. We carry out a systematic investigation of the strict notion of betweenness that Huntington employs (the three arguments are supposed to be distinct) by comparing it to McPhee's three axiom systems for the same intended class of structures, which employs weak betweenness (the arguments are permitted to be equal). Upon closely inspecting the proof that McPhee's axiom systems are equivalent to Huntington's (subject of course to the definition of weak betweenness in terms of strict and vice versa), one finds surprisingly that McPhee's axiom systems have quite different relations to strict betweenness.