A further simplification of Tarski's axioms of geometry

TL;DR

This paper proposes a slight modification to Tarski's axioms of geometry, enabling the omission and proof of an axiom, thereby simplifying the axiom system while preserving its independence properties.

Contribution

It introduces a modified axiom system for Euclidean geometry that reduces the number of axioms without losing logical independence and modularity.

Findings

One axiom is shown to be derivable in the new system.

The modified system retains all known independence properties.

Another axiom is proven independent in the new system.

Abstract

A slight modification to one of Tarski's axioms of plane Euclidean geometry is proposed. This modification allows another of the axioms to be omitted from the set of axioms and proven as a theorem. This change to the system of axioms simplifies the system as a whole, without sacrificing the useful modularity of some of its axioms. The new system is shown to possess all of the known independence properties of the system on which it was based; in addition, another of the axioms is shown to be independent in the new system.