Axiom:Euclid's Axiom

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Axiom

First Form

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

Let $=$ be the relation of equality.

This axiom asserts that:

$\forall a, b, c, d, t: \exists x, y:$

$\left({\mathsf{B}adt \land \mathsf{B}bdc \land \neg (a = d)}\right) \implies \left({\mathsf{B}abx \land \mathsf{B}acy \land \mathsf{B}xty}\right)$

where $a, b, c, d, t, x, y$ are points.

Intuition

Draw an angle $bac$ such that $0^{\circ} \le \angle{bac} \le 180^{\circ}$

Let $t$ be some point in the interior of $\angle{bac}$.

Draw a ray starting at $a$ and passing through $t$.

Let $d$ be a point on ray $at$ such that $d$ is between $a$ and $t$.

Then there is some line $x y$ passing through $t$, where $x$ is a point on one side of angle $bac$ and $y$ is a point on the other.

Second Form

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Source of Name

This entry was named for Euclid.

Also see

• Euclid's Fifth Postulate, an analogue of Euclid's Axiom 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. $183$ : Axiom $10$
  Illustration courtesy of Steven Givant.