- Incidence between points and lines:
- There are at least two distinct points.
- There is one and only one line that contains two distinct points.
- Every line contains at least two distinct points.

- Incidence between points and planes:
- There are three points that do not all lie on the same line.
- For any three points that do not lie on the same line there is a one and only one plane that contains them.
- Any plane contains at least three points.

- Incidence between lines and planes:
- If a line lies on a plane then every point contained in the line lies on that plane.
- If a line contains two points which lie on a plane then the line lies on the plane.

- Dimensionality of space:
- If two planes both contain a point then they also contain a line.
- 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