Log-euclidean geometry and "Grundlagen der Geometrie"

TL;DR

This paper introduces the simplest log-euclidean geometry, revealing a missing axiom in Hilbert's axiomatic system, and provides an elementary proof of the independence of certain axioms without Riemannian geometry.

Contribution

It defines a basic log-euclidean geometry and demonstrates the necessity of an additional axiom in Hilbert's axiomatic framework for geometry.

Findings

Log-euclidean geometry satisfies all but one of Hilbert's axioms.

Identifies a missing axiom in Hilbert's list related to congruence and parallels.

Provides an elementary proof of the independence of the missing axiom.

Abstract

We define the simplest log-euclidean geometry. This geometry exposes a difficulty hidden in Hilbert's list of axioms presented in his "Grundlagen der Geometrie". The list of axioms appears to be incomplete if the foundations of geometry are to be independent of set theory, as Hilbert intended. In that case we need to add a missing axiom. Log-euclidean geometry satisfies all axioms but the missing one, the fifth axiom of congruence and Euclid's axiom of parallels. This gives an elementary proof (with no need of Riemannian geometry) of the independence of these axioms from the others.