Axiom:Identity of Equidistance

Axiom

Let $\equiv$ be the relation of equidistance.

Let $=$ be the relation of equality.

Then the following axiom is imposed:

$\forall a, b, c: ab \equiv cc \implies a = b$

where $a, b, c$ are points.

Intuition

If two points have no distance between them, they are the same point.

Also see

• Identity of Points, an analogue of Identity of Equidistance in the context of Second-Order Logic.

Sources

• June 1999: Alfred Tarski and Steven Givant: Tarski's System of Geometry (Bull. Symb. Log. Vol. 5, no. 2: pp. 175 – 214) : p. $177$ : Axiom $3$