Can we alter the axioms of Euclidean space to have $\mathbb{Q}^3$ as a unique model? I posted this question at math.stackexchange.com but didn't get an answer.
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, to have $\mathbb{Q}^3$ as a unique model?
The crucial 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?
 A: Every consistent first-order axiomatic system has models of every cardinality. Given this fact, if your modified axioms has ${\mathbb{Q}}^3$ as a model, then ${\mathbb{Q}}^3$ will not be a unique model. However it still makes sense to ask if ${\mathbb{Q}}^3$ is a unique model.
So long as your modification uses only countably many symbols, then there will always be a countable models. Countable models are "discrete" in a sense.
In Hilbert's original writings, he came up with a concrete countable model to his axiomatization of Euclidean geometry. This model is the smallest Pythaogrean field containing ${\mathbb{Q}}$, i.e. $\sqrt{a^2+b^2}$ is in the field if $a$ and $b$ are. If you allow adding irrational points, this countable model is an instance of a discrete model.
A: As Colin Tan said, "[using] only countably many symbols, then there will always be a countable models."  Whereas the field $\mathbb{Q}$ of rational numbers uses a finite-number of symbols for an uncountable number of rational numbers.  I misunderstood and misapplied a concept.
The field generated by rational numbers is quite different from the "approximation space" rendered by using a finite number of bits interpreted as a floating-point number low-precision approximation to real numbers.  I'm editing my answer to point out my misunderstanding. @Hans-Stricker, I've fixed my error by pointing it out, but leaving it up (below the ruled line) so that some other bit-flipper like me will see why {0,1}$^n \times${0,1}$^n$ is not equivalent to $\mathbb{Q}$
below this is my original (erroneous) answer

Similarly, every numerical simulation in physics (or chemistry, biology, physiology, or medicine) always has to use finite precision representation of values, such that there is a limit to the largest and smallest integer represented by a fixed number of bits, and such that there is a limited amount of "floating-point-precision" available in dividing the bits of a floating-point representation of a real number into a set number of bits for the mantissa and a set number of bits for the exponent.
For example, assuming that $d=64$-bits are used to represent "real numbers" as floating point numbers in computations, $m=48$ bits may be allocated to the mantissa, allowing the numerator to be $2^{48}$ yielding approximately $14$ digits of base-ten specificity to the numerator; this leaves $d-m=16$ bits to the exponent which may be signed (+/-) yielding a range of -32768 to +32767.
In this case, the floating point number is in the range $n\times 2^{d-m}$, where $-(2^{47} \le n \le +(2^{47}-1)$, and ${-32768} \le d \le {32768}$.
If the total number of bits is $d$, the number of bits allocated to the exponent, $m$, may be decreased while simultaneously increasing the number of bits, $d-m$, allocated to the mantissa, increasing the "precision" of the numerator while decreasing the range over $\mathbb{R}$ spanned by this particular approximating set of {0,1}$^m \times ${0,1}$^{d-m}$ (which is not equivalent to $\mathbb{Q}$, as I erroneously stated originally)
Thus every numerical simulation is already, in a way, based on $\mathbb{Q}^d$ when models of $d$-dimensional systems are created and iterated using Euler or Runge-Kutta of whatever order.  
