Assuming that we are building our geometry on the axioms of Euclid/Hilbert, and using either the Dedeking or Cauchy construction of the reals, how can one prove this statement? I've looked up on the Internet, but half of the sources says it is an axiom, the other half defines a straight line as the set of the real solutions of a linear equation right in the beginning.
I know that the set $\mathbb{R}$ was created for the purpose of creating such a bijection, and consequently another bijection from $\mathbb{R}^2/\mathbb{R}^3$ and the points of plane/space, thus uniting algebra and geometry, but once the axioms of Euclidean geometry are listed and the reals constructed, a proof seems necessary to me, unless the way to go is to understand the concept intuitively, get rid of synthetic geometry, and define plane/space so that it will automatically coincide with $\mathbb{R}^2/\mathbb{R}^3$.
Is this the case? Or you can write a proof/give a link to one? I realize I might have written a complete mess, but I'm still a high school student and my only-basic English skills do not help.
Thank you for any answer!