<sup>I posted this question at [math.stackexchange.com][4] but didn't get an answer.</sup>

##Motivation##

Physicists are in search for a model of discrete space(-time) for a long time. So I wondered why not start with a "somehow discrete" space? How far do we get? 

## Question ##

Can we alter the axioms of Euclidean space, e.g. [Hilbert's axioms][1], to have $\mathbb{Q}^3$ as a unique model?

The crucial axioms seem to be the [congruence axioms][2] IV.1 and IV.4, and presumably the [line completeness axiom][3] 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?



  [1]: http://en.wikipedia.org/wiki/Hilbert%27s_axioms
  [2]: http://en.wikipedia.org/wiki/Hilbert%27s_axioms#IV._Congruence
  [3]: http://en.wikipedia.org/wiki/Hilbert%27s_axioms#V._Continuity
  [4]: http://math.stackexchange.com/questions/12750/can-we-alter-hilberts-axioms-to-have-mathbbq3-as-a-unique-model