We present an elementary system of axioms for the geometry of Minkowski spacetime. It strikes a balance between a simple and streamlined set of axioms and the attempt to give a direct formalization in first-order logic of the standard account of Minkowski spacetime in [Maudlin 2012] and [Malament, unpublished]. It is intended for future use in the formalization of physical theories in Minkowski spacetime. The choice of primitives is in the spirit of [Tarski 1959]: a predicate of betwenness and a four place predicate to compare the square of the relativistic intervals. Minkowski spacetime is described as a four dimensional 'vector space' that can be decomposed everywhere into a spacelike hyperplane-which obeys the Euclidean axioms in [Tarski and Givant, 1999]-and an orthogonal timelike line. The length of other 'vectors' are calculated according to Pythagoras' theorem. We conclude with a Representation Theorem relating models M of our system M 1 that satisfy second order continuity to the mathematical structure xR 4 , η ab y, called 'Minkowski spacetime' in physics textbooks. * We are grateful to John Burgess, Dino Calosi, Harold Hodes and Chris Wüthrich for discussion and comments on parts of this paper.

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Modal relationism is the view that our best physical theories can dispense with substantival space or spacetime in favor of possible configurations of particles. Kenneth Manders argued that the substantivalist conception is equivalent to this Leibnizian conception of space. To do so, Manders provides a translation f from the Newtonian theory T N into the Leibnizian modal relationist account T L . In this paper, we show that the translation does not establish equivalence, since there is no translation g : T L → T N that preserves theoremhood. This seems to show that the modal relationist theory T L is less parsimonious than its substantivalist rival.