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We define the simplest log-euclidean geometry. This geometry exposes adifficulty hidden in Hilbert's list of axioms presented in his "Grundlagen derGeometrie". The list of axioms appears to be incomplete if the foundations ofgeometry are to be independent of set theory, as Hilbert intended. In that casewe need to add a missing axiom. Log-euclidean geometry satisfies all axioms butthe 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 theindependence of these axioms from the others.

Log-euclidean geometry and "Grundlagen der Geometrie"