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In KG Binmore's "Topological Ideas" he says

The geometric terms which appear in Hilbert's axioms are the words point, line, lie on, between and congruent. To show $\mathbb{R}^2$ is a model for Euclidean plane geometry one has to give a precise definition of each of these words in terms of $\mathbb{R}^2$ and then prove each of Hilbert's axioms for Euclidean plane geometry as a theorem in $\mathbb{R}^2$... Interested readers ill find the book Elementary geometry from an advanced standpoint by E. E. Moise (Addison-Wesley, 1963) an excellent reference.

Except I recently got this book and it does not do this. It is an interesting book, but it simply accepts the primitive notions ("the geometric terms") and Hilbert's axioms. I would like to see the construction of these entities directly from $\mathbb{R}^n$, Binmore defines lines, circles, points, and planes. But nowhere is there congruence (presumably for line segments this would be the usual distance between two points) or betweenness defined (presumably a point $b$ would be between $a$ and $c$ if $d(a,c) = d(a,b) + d(b,c))$. I've attempted to do this myself but I'm in over my head a little bit.

So, does anyone have resources that systematically defines each of these geometric objects as sets of $\mathbb{R}^n$ and then proves hilbert's axioms as theorems in $\mathbb{R}^n$? Especially Euclid's postulate.

Andrews
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Remus Bleys
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  • You might want to look at chapter IV of "Foundations of geometry" by Karol Borsuk and Wanda Szmielew (1960). They prove that $\mathbb{R}^3$ is a model for Hilbert's axioms for Euclidean 3D geometry.(the axioms are slightly modified compared to original Hilbert's work) – Kulisty Jan 21 '19 at 19:31

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What you have is basically real linear algebra with usual metric. Define a point as an element of $\Bbb R^n$, a line as the set $\vec{a} + t\vec{b}$, betweeness exactly like you did (using distances - in this case, the norm), and planes and other hyperplanes as you'd do with subspaces on $\Bbb R^n$.

You’ll might be interested in this question.

Lucas Henrique
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  • So, I get that you would define two lines to be parallel if the intersection is zero, how do you show that given a point and a line on a plane there is one and only one parallel line containing that point?

    How would you define angles? Dot products?

     – Remus Bleys Jan 19 '19 at 01:54
  • Angles are actually not cited by Hilber, but you can use angles from the dot product definition and Cauchy-Schwarz, while $\cos$ and $\sin$ being solely algebraic functions (in the sense that they don't depend on geometric reasoning). Dot product is usual coordinate product and sum. Say that a plane is a subspace of dimension 2; for a line on a plane spanned by $\vec{u}, \vec{v}$ you can do a change of basis to get this plane to $xy$ (in terms of Cartesian coordinates) and then the proof is very simple. – Lucas Henrique Jan 19 '19 at 02:02