Axiom:Axiom of Segment Construction

Axiom

Let $\equiv$ be the relation of equidistance.

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

The axiom of segment construction is given by the following statement:

$\forall a, b, c, q: \exists x: \mathsf{B} qax \land ax \equiv bc$

where $a, b, c, q, x$ are points.

Intuition

Let $bc$ be a line segment.

Let $a$ be a point on a ray with endpoint $q$.

One can create a line segment congruent to $bc$. This line segment is $ax$.

Sources

• June 1999: Alfred Tarski and Steven Givant: Tarski's System of Geometry (Bull. Symb. Log. Vol. 5, no. 2: pp. 175 – 214) : p. $178$ : Axiom $4$
  Illustration courtesy of Steven Givant.