Question

Prove that the axiomatic set for Fano's 7 point/line geometry is independent

Answer 1

Axioms for Fano's Geometry

Undefined Terms. point, line, and incident.
Axiom 1. There exists at least one line.
Axiom 2. Every line has exactly three points incident to it.
Axiom 3. Not all points are incident to the same line.
Axiom 4. There is exactly one line incident with any two distinct points.
Axiom 5. There is at least one point incident with any two distinct lines.

Fano's Theorem 1. Two distinct lines intersect in exactly one point.

Proof. Let p and q be any two distinct lines. By Axiom 5, there is a point A incident to both p and q. Suppose there is a second point B, distinct from A, incident to both pand q. Then by Axiom 4, p and q are the same line, but this contradicts that p and qare distinct lines. Thus p and q intersect in exactly one point A. Therefore, two distinct lines intersect in exactly one point.

lines intersect in exactly one point.// [Link to lecture on the proof of Fano's Theorem 2.]

Fano's Theorem 2. Fano's geometry consists of exactly seven points.

Proof. By Axiom 1, there exists a line l. Then by Axiom 2, there exist exactly three points A, B, C on line l. Now by Axiom 3, there exists a point P not on line l. Hence we have at least four distinct points A, B, C,and P. By Axiom 4 and since P is not on line l, there are three distinct lines AP, BP,and CP. And by Axiom 2, each of these lines contains a third point D, E, and F on AP, BP,and CP, respectively. None of D, E or F can be any of the points A, B, C, or P; for if this was not true, Axiom 4 would be contradicted.Hence there are at least seven distinct points A, B, C, D, E, F,and P.  

We assert that there are exactly seven distinct points. Suppose there exists a distinct eighth point Q. Note Q is not on l,since A, B, and C are the only points on l. By Axioms 4 and 5, lines PQ and l must intersect at a point R. Since A, B, and C are the only points on l, R must be one of A, B, or C. Suppose R = A. Since D is on line APand A = R is on line PQ, we would have R = A, D, P, and Q collinear which contradicts Axiom 2. The other cases for B or C are similar. Hence there are exactly seven distinct points.