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In Apostol's Mathematical Analysis (second edition), it is written, on p.3:

The real numbers are often represented geometrically as points on a line (called the real line or the real axis). A point is selected to represent $0$ and another to represent $1$ [...]. This choice determines the scale. Under an appropriate set of axioms for Euclidean geometry, each point on the real line corresponds to one and only one real number and, conversely, each real number is represented by one and only one point on the line. It is customary to refer to the point $x$ rather than the point representing the real number $x$.

Question: Is there a formal proof of the bold sentence? If so, where can I find such a proof?

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  • $\begingroup$ I'm not absolutely sure it's what he means, but you can reconstruct all of euclidian geometry by taking (for example) the plane to be the set $\Bbb R^2$. That way you can prove Euclid's axioms in the framework of set theory. Have a look at Affine space on Wikipedia. Once the axioms are proved, you can do good old "synthetic" geometry if you like, or use analytic or vector tools, etc. $\endgroup$ Commented Nov 12, 2015 at 15:01
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    $\begingroup$ Find Hilbert's axioms for Euclidean geometry. Using them, you can construct operations on a line (with 0 and 1 chosen as noted) so that you get a complete ordered field. This is probably even in Hilbert's original paper. $\endgroup$
    – GEdgar
    Commented Nov 12, 2015 at 15:05
  • $\begingroup$ @GEdgar Ah, this kind of stuff! Then Artin's "Geometric Algebra" has this too. $\endgroup$ Commented Nov 12, 2015 at 15:14
  • $\begingroup$ This reminds me of the Cantor-Dedekind axiom books.google.com/… $\endgroup$
    – iMath
    Commented Nov 11, 2017 at 8:15

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