Equivalency of Pasch's Postulate and the plane separation postulate

by: Robert Hansen, proud member of the math squad.

 keyword: tutorial, geometry 

Introduction

Consider the following two statements:

1) If a line does not pass through a vertex of a triangle, it must pass through either exactly 0 or exactly 2 of its sides

2) Given a line l and a plane P containing it, one can find two distinct sets A and B such that the union of A, B, and l is P, A, B and l are disjoint, A and B are convex and given a point in A and a point in B, the segment connecting them must intersect l.

These two statements are Pasch's Postulate and The Plane Separation Postulate, respectively. They are equivalent, and one can construct an if and only if relationship between them. Furthermore, they are both unprovable from more basic axioms, and are rather famous for showing the incompleteness of Euclid's axioms.

Background

Let's talk about some axioms and definitions we'll need to do this proof:

Axiom 1: Through any two different points, there exists exactly one line.

Axiom 2: Through any three non-colineear points there exists exactly one plane.