Axiom:Inner Transitivity of Betweenness

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.

Axiom

Let $\mathsf{B}$ be the relation of betweenness.

This axiom asserts that:

$\forall a, b, c, d: \mathsf{B}abd \land \mathsf{B}bcd \implies \mathsf{B}abc$

where $a, b, c, d$ are points.

Intuition

Let $abd$ and $bcd$ be line segments.

Suppose they are arranged to form one straight line segment.

Then the points $a, b, c$ are collinear.

Note that this axiom still holds in the degenerate cases where the points are not (pairwise) distinct.

For example, if we are dealing with exactly three points, this axiom could be interpreted as "three points on a line are collinear".

Also see

• Outer Transitivity of Betweenness

Sources

• June 1999: Alfred Tarski and Steven Givant: Tarski's System of Geometry (Bull. Symb. Log. Vol. 5, no. 2: pp. 175 – 214) : p. $185$ : Axiom $15$