Following this paper, we can define a straight line as a triplet $(L,\mathcal{B},\cong)$ were $L$ is a set (the set of points in the line), $\mathcal{B}$ is a ternary relation over $L$ (called the betweenness relation), $\cong$ is a binary relation (called the congruency relation) over the segments of $L$ and the following series of betweenness and congruency postulates hold:
- Given three points, at most one of them is between the other two.
- Given two points, there is at least one additional point between them.
- If $C$ is between $A$ and $B$, $D$ is between $A$ and $B$ and $C\not=D$, then $D$ is between $A$ and $C$, or between $B$ and $C$.
From now on, if $C$ is between $A$ and $B$, we will write $A$-$C$-$B$.
- If $A$-$B$-$C$ and $A$-$D$-$C$, then $A$-$D$-$B$.
- If $A$-$B$-$C$ and $C$-$B$-$D$, then $A$-$C$-$D$.
Note: The segment determined by $A$ and $B$ is the set \begin{equation}\overline{AB}:=\{A, B\}\cup\{C\in L:\ A\text{-}C\text{-}B\}\text{. }\end{equation}
- If $\overline{AB}\cong\overline{CD}$ and $\overline{EF}\cong\overline{CD}$, then $\overline{AB}\cong\overline{EF}$.
- Given $\overline{AB}$ and $\overline{CD}$ two non-degenerate segments, there is an unique point $E$ such that $\overline{CD}\cong\overline{AE}$ and $A$-$E$-$B$ or $A$-$B$-$E$.
- If $A$-$C$-$B$, $A'$-$C'$-$B'$, $\overline{AC}\cong\overline{A'C'}$ and $\overline{BC}\cong\overline{B'C'}$, then $\overline{AB}\cong\overline{BC}$.
Also, the following Dedekinds postulate hold:
- Given a segment $\overline{AB}$ and two subsets $M$ and $N$ of $\overline{AB}$ such that if $C\in M$ and $D\in N$ then $\overline{AC}\subset\overline{AD}$, then there is a point $E$ such that $\overline{AC}\subset\overline{AE}\subset\overline{AD}$ for all $C\in M$ and $D\in N$.
Given this purely synthetic structure (or another one similar), how can we define the real number algebraic structure, and, as a side effect, a coordinate system over a straight line.
I'm not asking for a fully detailed and painful proof, just for an sketch or an idea.
But why on Earth would you want to do that?
In the context of a metric approach to geometry with a Birkhoff-like (metric) axiom system, the introduction of the ruler postulate seems a bit artificial to me. After all, why on earth the real numbers ''are'' a line? So, giving a synthetic axiomatic system for a line, and proving that "it matches" with $\mathbb{R}$ will solve this issue.