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^{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?

^{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?

^{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?

^{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?

^{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 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?

^{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?