Axiom:Pasch's Axiom (Tarski's Axioms)

This page is about Pasch's Axiom in Tarski's Geometry. For other uses, see Axiom:Pasch's Axiom.

Axiom

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

First form

The first form of the axiom is:

$\forall a, b, c, p, q: \exists x :\mathsf{B}apc \land \mathsf{B}bqc \implies \mathsf{B}pxb \land \mathsf{B}qxa$

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

Intuition

Let $aqc$ be a triangle.

Draw a line segment extending segment $cq$ to some point $b$ outside the triangle such that $c, q, b$ are collinear.

Pick a point $p$ on segment $ac$.

Draw a line segment connecting point $p$ with point $b$.

Segment $pb$ will intersect segment $aq$ at some point $x$.

Second form

The second form of the axiom is:

$\forall a,b,c,p,q : \exists x : \mathsf{B}apc \land \mathsf{B}qcb \implies \mathsf{B}axq \land \mathsf{B}bpx$

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

Intuition

Let $a, p, c$ be collinear.

Further, let $q,c,b$ be collinear.

Construct a ray with endpoint $a$ passing through $q$.

Construct another ray with endpoint $b$ passing through $p$.

Ray $aq$ and ray $bp$ will intersect at some point $x$.

Also see

• Equivalence of Formulations of Pasch's Axiom

Source of Name

This entry was named for Moritz Pasch.

Sources

• June 1999:  Alfred Tarski and Steven Givant: Tarski's System of Geometry (The Bulletin of Symbolic Logic Vol. 5, no. 2: 175 – 214) : Page 179, 180 : Axiom $7$
  Illustration courtesy of Steven Givant.