Question: describe a finite model such that no two lines have...
Describe a finite model such that no two lines have the same set of incident points, axioms I2 and I3 are true, but I1 is false.
Can you give a model as above with the additional constraint that for any two points, there are at least two lines incident to both?
I1: For any two points, there is a unique line incident to both.
I2: For every line, there are at least two points incident to it.
I3: There exist three non-collinear points.
Hints: It is doable with 4 points. 3 of them might be collinear. Pick some subsets of points cleverly and declare them to be lines. Make sure you don’t ruin I3 in the meantime.