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Log-euclidean geometry and "Grundlagen der Geometrie"

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.
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https://hal.archives-ouvertes.fr/hal-02369425
Contributor : Ricardo Pérez-Marco Connect in order to contact the contributor
Submitted on : Tuesday, November 19, 2019 - 2:56:47 AM
Last modification on : Sunday, June 26, 2022 - 9:48:44 AM
Long-term archiving on: : Thursday, February 20, 2020 - 3:45:04 PM

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  • HAL Id : hal-02369425, version 1
  • ARXIV : 1911.08919

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Ricardo Pérez-Marco. Log-euclidean geometry and "Grundlagen der Geometrie". 2019. ⟨hal-02369425⟩

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