Proposition 24 is For every point there is at least one line

Proposition 2.4 is For every point, there is at least one line not passing throught it.

Proof: According to Axiom I-3, there are three points (call them A, B, and C) such that no line is incident with all of them. Let P be A. Then P does not lie on BC.

Why is this proof not correct.

Solution

AXIOM: There exists three distinct points not simultaneously incident with a common line l. PROPOSITION ; There exists three distinct lines not through any common point.

PROOF: let l,m,n be three distinct lines not concurrent by above preposition . given any arbitrary point P, then one of l,m,n doesn\'t pass through P