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Finite Axiomatic Geometry for Undefined Terms

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Consider the following axiom system.
The undefined terms are point, line, and on.
Axioms:
I. Given any two distinct points, there exactly one line on both of them.
II. Given any line, there is at least one point not on it.
III. Given any line, there are at least five points on it.
IV. There is at least one line.

Question:
(a) Prove that there are at least six lines.

(b) Prove that there are at least 21 points.

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Solution Summary

Finite axiomatic geometry for undefined terms is investigated and discussed with details in the solution.

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Proof:

(a) According to IV, there is at least one line, say L. According to III, there are at least five points A,B,C,D,E on the line L. According to II, there is at least one point, say F, not on L. Now according to I, A and F are two ...

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