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.