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Judit, X. Madárasz and István, Németi and Gergely, Székely (2005) Twin Paradox and the Logical Foundation of Relativity Theory.

http://philsci-archive.pitt.edu/archive/00002358/

They seem to be saying that inertial frames can be dealt with using algebraic numbers, which allows a 1st order axiomatization. However accelerated frames require further axioms, equivalent to moving to the real numbers (I'm not sure whether this means higher order axioms). This seems reasonable to me, since acceleration means differential equations, which means going beyond the algebraic numbers.

I'd be interested to hear other people's opinions on this.