Axiom:Density of Betweenness

Axiom

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

Let $=$ be the relation of equality.

Then the density of betweenness is expressed by the statement:

$\forall a,b: \exists c: \neg \left({a = b}\right) \implies \neg \left({c = a}\right)\land \neg \left({c = b}\right) \land \mathsf{B}acb$

where $a,b,c$ are points.

Intuition

Between any two distinct points you can draw a third point, distinct from both.

Sources

• June 1999:  Alfred Tarski and Steven Givant: Tarski's System of Geometry (The Bulletin of Symbolic Logic Vol. 5, no. 2: 175 – 214) : Page 188 : Axiom $22$