Problem 1: What Is A Geometry?

Index

For this course we will define the objects in our geometry as a non empty set of points and a non empty set of lines, where a line is a given subset of the set of points that contains at least two elements. Geometries can be either finite or infinite depending on whether they have a finite or infinite number of points, but for now, we will be looking mostly at finite geometries.

Furthermore, the objects in a given geometry are defined by a set of axioms. Any constructed example of the geometry must satisfy each and every axiom.

We will look at the automorphims of a given geometry in a later problem.

Before we go any further, let's take a look at an example of a geometry.

NEAR LINEAR SPACES Any line has at least two points. Two points are on at most one line.

Here is an example of this geometry:

Can you construct others?

This is a very important type of geometry. In addition, this broad geometry contains many other more restricted and complicated geometries within it (compare with Klein's heirachy of geometries). We'll see some examples soon.

Using this example, lets now define other terms that we use to describe the axiom systems. We will call an axiom system consistent if we are able to construct an example that satisfies our axioms. In other words, a consistent axiom system must be one such that no one axiom negates the existence of another. If no such example exists then our set of axioms will be called inconsistent. Our set of axioms above are obviously consistent, as we were able to construct an example that satisfied them.

PROBLEM 1: (a) Find an example of an inconsistent set of axioms for a geometry. (b) Find an example of a consistent geometry with at least three axioms. Solutions

References: Batten, Bennet, and Polster.