The Axioms of Incidence

The following axioms set out the basic incidence relations between lines, points and planes. They also characterise the concept of ``dimension'' that we associate with these notions.

1. Incidence between points and lines:
   1. There are at least two distinct points.
   2. There is one and only one line that contains two distinct points.
   3. Every line contains at least two distinct points.
2. Incidence between points and planes:
   1. There are three points that do not all lie on the same line.
   2. For any three points that do not lie on the same line there is a one and only one plane that contains them.
   3. Any plane contains at least three points.
3. Incidence between lines and planes:
   1. If a line lies on a plane then every point contained in the line lies on that plane.
   2. If a line contains two points which lie on a plane then the line lies on the plane.
4. Dimensionality of space:
   1. If two planes both contain a point then they also contain a line.
   2. There are at least four points that do not all lie on the same plane.

The first four axioms (which do not refer to planes) are called the plane geometry axioms, while the remaining are the space axioms. Out of the various Theorems that can be proved we note

Theorem 1   Given a line and a point not on it there is one and only one plane that contains the line and the point.

Theorem 2   Given a pair of lines which meet in a point there is one and only one plane that contains the lines.

Theorem 3   Given four points that do not all lie on a plane, there is no line containing three of these points.

Exercise 1   There is a ``geometry'' consisting of 4 points, 6 lines and 4 planes that satisfies these axioms.

Exercise 2   Which of the above axioms can be omitted? For those that are necessary construct a ``geometry'' that satisfies the chosen axiom and defies the others.