What properties of numbers are needed to model accelerated observers in relativity?

We investigate the possible structures of numbers (as physical quantities) over which accelerated observers can be modeled in special relativity. We present a general axiomatic theory of accelerated observers which has a model over every real closed field. We also show that, if we would like to model certain accelerated observers, then not every real closed field is suitable, e.g., uniformly accelerated observers cannot be modeled over the field of real algebraic numbers. Consequently, the class of fields over which uniform acceleration can be investigated is not axiomatizable in the language of ordered fields.