Axiom:Identity of Betweenness

Axiom

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

Let $=$ be the relation of equality.

Then the following axiom holds:

$\forall a,b: \mathsf{B}aba \implies a = b$

where $a$ and $b$ are points.

Intuition

If between a point and itself lies another point, we're dealing with exactly one point.

Sources

• June 1999: Alfred Tarski and Steven Givant: Tarski's System of Geometry (Bull. Symb. Log. Vol. 5, no. 2: pp. 175 – 214) : p. $179$ : Axiom $6$