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It has been shown that the Euclidean plane defined by Hilbert's axioms is isomorphic to the 2D Euclidean vector space. Spacetime in special relativity can't be modeled by an Euclidean vector space, so not all of Hilbert's axioms can hold in the geometry of special relativity.

My question is: which of them are violated and which of them still hold?

And going one-step further: can the axioms which are violated be replaced by others so that we get a synthetic version of the geometry of special relativity analogous to axiomatic Euclidean geometry?

Any thoughts and references are appreciated. I didn't post this question under physics because I don't think that many physicists are familiar with axiomatic geometry.

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    $\begingroup$ I don't know much about this, but I found this on Google Books: books.google.com/books?id=avy6BQAAQBAJ&pg=PA40&lpg=PA40 $\endgroup$ – David Jan 13 '16 at 12:22
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    $\begingroup$ There's a chapter on special relativity in Efimov's Higher Geometry, a book that also has extensive discussion of axiomatic issues in geometry. $\endgroup$ – David Jan 13 '16 at 12:41
  • $\begingroup$ I just skimmed the relevant chapter of Efimov's book (wich is quite hard to find by the way) and although he does discuss axiomatics at length his treatment of the geometry of special relativity uses vector space language exclusively. Thanks anyway, David! $\endgroup$ – Marc Jan 14 '16 at 15:49
  • $\begingroup$ All right. Sorry it didn't have what you wanted. $\endgroup$ – David Jan 14 '16 at 18:23

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