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.
    • $f_r(x)=(1+|a|) x$.

\tikzmath{\Size=1.5;};
\tikzmath{\Angle=30;};


\input{Defs.tikz}
\input{1.tikz}

\draw (Text) node [] {$(1)$};

\input{Inc.tikz}


\input{1.tikz}
\input{2.tikz}

\draw (Text) node [] {$(2)$};


\input{Inc.tikz}


\input{1.tikz}
\input{2.tikz}
\input{3.tikz}

\draw (Text) node [] {$(3)$};


\input{Inc.tikz}


\input{1.tikz}
\input{2.tikz}
\input{3.tikz}
\input{4.tikz}

\draw (Text) node [] {$(4)$};

\draw[-Latex]
   ($(A)!\ta!(B)$)
   --
   ($(B)!\ta!(A)$)
   node [right] {$r$};


\draw[dashed,-Latex]
   ($(a)!\ta!(b)$)
   --
   ($(b)!\ta!(a)$)
   node [right] {$\mathbb{R}$};

\filldraw (A) circle (1pt) node [above] {$A$};
\filldraw (B) circle (1pt) node [above] {$B$};
\filldraw (a) circle (1pt) node [below] {$a$};
\filldraw (b) circle (1pt) node [below] {$b$};