Critique the given proof of the following statement in Incid

Critique the given proof of the following statement (in Incidence Geometry): Statement: For any given point, there exists a line passing through that point. Proof: By axiom I1, there exists a line 1. By axiom I2, 1 contains at least two distinct points (A and B). Therefore, for any point, there exists unique line that is incident with it.

Solution

The axiom I3 precisely says that there exists three non-collinear points on a plane.

To prove that there exists a line that passing through a given point, first we need to consider two points on the plane. Let A be the given point, consider another point say B on the same plane. Then by axiom I1, there exists a unique line that contains the points A and B. Let the line be l. Therefore, the line l is a line passing through the point A.

But note that this line is not unique as far as only the point A is concerned. If we consider another point C instead of B on the same plane and the unique line joining A and C, that line will still pass through A, but will be different from l.

The given proof does not use the third axiom in chooisng another point and uses the second axiom to find a point on the line l which isnot necessary. Also, the uniqueness stated in the proof is also not correct.