Can we alter Hilbert's axioms to have $\mathbb{Q}^3$ as a unique model?
The critical axioms seem to be the congruence axioms IV.1 and IV.4, and presumably the line completeness axiom V.2.
But how are they to be modified?
IV.1 might be replaced by requiring that there are counter-examples (irrationality of $\sqrt{2}$) and appropriately relaxing "congruent" to "almost congruent" (= "arbitrarily close to congruence").
But what about line completeness then, since it might be possible to add irrational points to $\mathbb{Q}^3$ such that the modified axioms still hold?