Disclaimer: I had probably used some kind of "extended" set of incidence axioms, which also includes planes. Even though this is part of my homework, I seriously doubt this is what I was expected to do. So, while I will try to find the exact set of incidence axioms my prof. meant, this is still a question that I'd like to have answered. Below is the set of axioms the way I found it here: https://www.imsc.res.in/~kapil/geometry/euclid/node2.html the axiom of parallels: https://www.imsc.res.in/~kapil/geometry/euclid/node4.html I will also repeat them for ease of reference:
- 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.
- 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.
- 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.
- 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 axiom of parallels:
- Given a line and a point outside it there is exactly one line through the given point which lies in the plane of the given line and point so that the two lines do not meet.
When I try to build the model, in a loop, I keep adding more planes (because of 5) and once I added a plane, it requires that there be four points in the plane (because of the axiom of parallels), then I get new planes (again because of 5) and the loop continues.
My homework asked whether it is necessary that such a system have 6 and then 7 lines, but it seems that it will only make sense to answer such question, if I never construct additional planes, or does it?
On the second thought, would a "cube" with lines connecting all of its points and planes drawn whenever necessary (this will give 12 planes and 36 lines) be such a model?