Simplifying the axiomatization for the order affine geometry

TL;DR

This paper simplifies the axiomatization of ordered affine geometry by replacing a complex convergence axiom with a more intuitive one, reducing redundancy and improving conceptual clarity.

Contribution

It introduces a new, simpler axiom (ODO) to replace the complex I.7 axiom, streamlining the axiomatization of ordered affine geometry.

Findings

Axiom I.7 is redundant and can be replaced

The new ODO axiom captures properties of oppositely directed lines

Potential replacement of I.6 with ODO is explored

Abstract

Based on an ordering with directed lines and using constructions instead of existential axioms, von Plato proposed a constructive axiomatization of ordered affine geometry. There are 22 axioms for the ordered affine geometry, of which the axiom I.7 is about the convergence of three lines (ignoring their directions). In this paper, we indicate that the axiom I.7 includes much redundancy, and demonstrate that the complicated axiom I.7 can be replaced with a simpler and more intuitive new axiom (called ODO) which describes the properties of oppositely and equally directed lines. We also investigate a possibility to replace the axiom I.6 with ODO.