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 (Bull. Symb. Log. Vol. 5, no. 2: pp. 175 – 214) : p. $188$ : Axiom $22$