Ruler Axiom

Exists Measure of Distance

• Axiom: $\forall r \in \mathcal{R}$, exists Bijective Distance Function, $f: r \mapsto \mathbb{R}$: \[ d(A,B)= \left| f_r(B)-f_r(A) \right| \]
• Definition, $X$ is inbetween $A$ and $B$:
  • $A, B, X \in \mathcal{P}$ colineares.
  • $d(A,X)+d(B,X)=d(A,B)$.
• Put: $a=f(A)$, $b=f(B)$, $x=f(X)$.
• $a*x*b \quad\Leftrightarrow\quad a<x<b \vee b<x<a$
• Teorema: \[ A*X*B \quad\Leftrightarrow\quad a*x*b \]
• Models and Distance Functions:
  • Cartesian:
    • Points: $\mathbb{R}^2$: $(x,y)$.
    • Lines: $y=ax+b$, $a,b \in \mathbb{R}$.
    • $f_r(x)=\sqrt{1+a^2} x$.
  • Taxist:
    • Points and Lines as in Cartesian.
    • $f_r(x)=(1+|a|) x$.