Finite Geometry Axiom system 1:

There exists a set of objects called points
There exists a collection of objects called lines, where each line is a subset of the set of points.

  1. For every two distinct points there is exactly one line that contains both of them
  2. Any time there is a line, and a point that is not on the line, then there is another line that contains the point, and does not intersect with the first line.
  3. There exist at least 4 points such that no three of those points are on the same line.
Points and Lines are commonly chosen to be undefined terms in modern geometry descriptions.