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Question: describe a finite model such that no two lines have...

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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.

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