Projective Plane Axioms:
- 1) A line lies on at least two points
- 2) Any two distinct points have exactly one line in common
- 3) Any two distinct lines have at least one point on common
- 4) There is a set of four distinct points, no three are collinear.
Proof: By PA4, there is a set of four distinct points, no three collinear. By definition of collinear this means that we can create a set of four points where no three lie on the same line. We know by PA3 that any two distinct lines have one point in common. Since no three points lie on the same line, the third point, P3, must lie on a second line, L2 different from the first line, L1, containing points P1 and P2. Then we must draw a third line, L3, connecting P3 to P1. Now, consider a fourth point, P4. We construct lines L4, L5 and L6 such that PA are met.
I'm not sure if I'm going about this the correct way or if there's a better way to think about this. Any help is appreciated. Thanks!