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I know that the Pythagorean already showed that the set of all rationals cannot measure all "real" distances, but what if we assume that the number of points and distances in Euclid plane is countable (but not rationals). In this case if we add all the irrational numbers obtained in by geometric figures (such as the hypotenuse of right triangles) we still remain with countable set of measures, provided that we forbid the sum of infinite distances... Now suppose that we do forbid the infinite sum of measures, do we still have to assume that the number of points (or distances) is continuous? Can we found other model (preferably countable), beside the $\mathbb{R}^2$ Cartesian model, that models the Euclidean plane?

Aloizio Macedo
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Yonatan
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  • I believe if you replace the reals with the real algebraic numbers, that's enough.

    https://en.wikipedia.org/wiki/Algebraic_number

     – Almentoe Oct 04 '15 at 15:44
  • Or, indeed, https://en.wikipedia.org/wiki/Constructible_number . – Chappers Oct 04 '15 at 15:46
  • For talking about lines, the algebraic numbers may be enough (I don't really know... for instance, it is not clear that all two lines must intersect or be parallel since the IVT will not hold anymore). But you'll most likely have trouble with circles, and circles are an interesting object in euclidean geometry. – Aloizio Macedo Oct 04 '15 at 15:49
  • @AloizioMacedo No, the algebraic numbers are a field (the sum, difference, product, and quotients of algebraically are algebraic), and the square root of an algebraic is algebraic. You can use this to show that you'll have no problem. (It's not obvious that they're a field — What polynomial does $\sqrt[3]2+\sqrt[3]3$ satisfy? — but they are.) – Akiva Weinberger Oct 04 '15 at 15:55
  • @AkivaWeinbergercolumbus Are you referring to the part where I wonder if two lines must always intersect (meaning, you are saying that they will)? If so, that's interesting. – Aloizio Macedo Oct 04 '15 at 15:56
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    @AloizioMacedo Yes. Two lines (assuming they each pass between two algebraic points) are either parallel or intersecting. Similarly for circles. – Akiva Weinberger Oct 04 '15 at 15:58

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A contructible number is a real number that can be obtained through integers, $+$, $-$, $\times$, $\div$, and square roots.

For example, $7+\sqrt{2-\sqrt3}-\sqrt{\sqrt{17}}$ is constructible. A famous theorem says $\sqrt[3]2$ is not constructible.

Let $C$ be the set of constructible numbers. Then $C^2$ should be a good model. It's the set of points obtained with straightedge and compass, where $(0,0)$ and $(0,1)$ are given.

Akiva Weinberger
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