Axiom:Hilbert's Axioms/Order

This article needs to be linked to other articles. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code.

Definition

Linear Axioms

\((\text {II}, 1)\)	$:$							If $A, B, C$ are points of a straight line and $B$ lies between $A$ and $C$, then $B$ lies also between $C$ and $A$.
\((\text {II}, 2)\)	$:$							If $A$ and $C$ are two points of a straight line, then there exists at least one point $B$ lying between $A$ and $C$ and at least one point $D$ so situated that $C$ lies between $A$ and $D$.
\((\text {II}, 3)\)	$:$							Of any three points situated on a straight line, there is always one and only one which lies between the other two.
\((\text {II}, 4)\)	$:$							Any four points $A, B, C, D$ of a straight line can always be so arranged that $B$ shall lie between $A$ and $C$ and also between $A$ and $D$, and, furthermore, that $C$ shall lie between $A$ and $D$ and also between $B$ and $D$

Plane Axiom

\((\text {II}, 5)\)

$:$

Let $A, B, C$ be three points not lying in the same straight line and let $a$ be a straight line lying in the plane $ABC$ and not passing through any of the points $A, B, C$. Then, if the straight line $a$ passes through a point of the segment $AB$, it will also pass through either a point of the segment $BC$ or a point of the segment $AC$.

Sources

• 1902: David Hilbert: The Foundations of Geometry (translated by E.J. Townsend): $\text I \S 3$. Group $2$: Axioms of order