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I am interested in reading a proof that the vector space $\mathbb{R^3}$ satisfies the axioms of an axiomatization of Euclidean geometry (such as Hilbert's), in which such theorems as Pythagoras' Theorem can be proven. I want to understand how concepts such as 'angle', 'plane' and 'area' from Euclidean geometry can be translated into $\mathbb{R^3}$ language.

I would also be interested to know whether there are axiomatizations of Euclidean geometry in an arbitrary finite number of dimensions.

What are some good resources which cover these type of questions? Is Elementary Geometry from an Advanced Standpoint by Moise what I am looking for?

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$\mathbb{R}^3$ is the three dimensional Euclidian space.

Planes are subsets obeying a relation of the form $ax + by + cz = d$ for which at least one of $a,b,c$ is nonzero.

Angles are defined by the formula $ \vec{x} \cdot \vec{y} = x_1 y_1 + x_2 y_2 + x_3 y_3 = |\vec{x}| |\vec{y}| \cos(\theta).$

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    $\begingroup$ As I currently understand it, $\mathbb{R}^3$ is a model for Euclidean space, while Euclidean space itself is the theory based axioms, those of Hilbert or Tarski for example. $\endgroup$ – Steven Wagter Jun 22 at 12:34
  • $\begingroup$ Well, you're right, but using the correct identifications of which I provided some, they are practically indistinguishable. $\endgroup$ – Alexander Geldhof Jun 23 at 7:56
  • $\begingroup$ I do not doubt that definitions such as yours are equivalent to one of the proper axiomatizations of Euclidean geometry. I want to be able to verify, and understand the equivalence, since your definitions aren't very elementary (they use real numbers, the vector space $\mathbb{R}^3$, the cosine function, Pythagoras' theorem). $\endgroup$ – Steven Wagter Jun 23 at 9:15
  • $\begingroup$ I'm sorry, I'm in the middle of exams right now, and while my math is usually still on point, reading fine print and correctly interpreting people isn't exactly my forte right now. Your original post is clear enough. Hopefully someone can answer this more thoroughly than I did. $\endgroup$ – Alexander Geldhof Jun 23 at 9:22
  • $\begingroup$ No worries, good luck on the exams. $\endgroup$ – Steven Wagter Jun 23 at 16:35
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You can define plane, angle, area, line, volume, curve, surface etc... first geometrically in $2$ and $3$ dimensions and then with the help of coordinates translate those terms in $\mathbb R^2$ and $\mathbb R^3$ in terms of equations satisfying some equalities or/and inequalities (line,plane,curve,surface) or in the terms of limits of sums (volume, area, length). All of geometrical intuition is helpful in $\mathbb R^2$ and $\mathbb R^3$ and, when you define distance from the origin of a point in $\mathbb R^2$ or $\mathbb R^3$ with the help of Pythagoras theorem then that distance is generalized to $\mathbb R^n$ as having exactly the same form as in $\mathbb R^2$ or $\mathbb R^3$ with only difference that it does not have $2$ or $3$ terms but $n$. Exactly the same is done with distance between any two points $x,y$. The properties of "distance function" are easily proven also in that general setting. Area and volume can be defined as set functions that have some properties we want them to have and then it is proved that integral of "nice" positive functions really is a good tool for defining area and volume in a sense that it has all the properties as area and volume defined as set functions. Angle is defined with a help of dot product and formula for dot product in $\mathbb R^n$ is straightforwardly generalized to have the same form as in $\mathbb R^2$ and $\mathbb R^3$.

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