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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?

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    @arbautjc: I wouldn't know, haven't read it. Seems to contain some interesting ideas though.2013-03-13

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In text of

'If A, B are two points on a line a, and if A′ is a point upon the same or another line a′ , then, upon a given side of A′ on the straight line a′ , we can always find a point B′ so that the segment AB is congruent to the segment A′B′'

it must be stated that a' is parallel or perpendicular to a, and the same one with angles. Archimedes axiom is still needed and the last one might be replaced with nearly opposite one:

A restriction of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follows from Axioms I-III and from V-1 is impossible.

Now we could take rational multiples of any segment on a line (Thet depends on similarity and maybe Pasch's axiom), and due to axiom of anti-completeness, no other segments are possible.