Axiom: "Between" Relation, Axioms of Order

Let $\mathcal P_l$ be a set of points that lie on a given straight line $l$. The between relation is a relation $R\subseteq \mathcal P_l\times \mathcal P_l\times \mathcal P_l$ fulfilling the following axioms:

Linear Axioms

If a point $B$ lies between a point $A$ and a point $C,$ then all these points are distinct and $B$ also lies between $C$ and $A.$


2 For any two points $A$, $C$ that lie on a straight line there is at least another point $B$ on that straight line such that $C$ lies between $A$ and $B$.


3 Of any three points that lie on a straight line, there is no more than one which lies between the other two.

Plane Axiom

4 Let $g$ be a straight line in a plane determined by three points $A$, $B,$ and $C$ such that none of these points lie on $g.$ If there is a point $D$ between $A$ and $B$ that lies on $g$, then there is also a point $E$ such that it lies on $g$ and also it lies either between $A$ and $C,$ or between $B$ and $C.$


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  1. Hilbert, David: "Grundlagen der Geometrie", Leipzig, B.G. Teubner, 1903