Consider the following axiom set: Undefined terms: point, line, plane, and incid

Question

Consider the following axiom set:

Undefined terms: point, line, plane, and incidence

Axioms:

(1) There exists at least two points.

(2) Given any two distinct points, there is a single unique line incident with botth.

(3) Every line contains exactly two points.

(4) For every line L there exists a point P that is not on L

(5) Given any three distinct non-colliniar points there exists a single uique plane A that contains them

(6) For every plane a there exists a point P that is not on A

I really need help with finding a model for this. but we also need to know how many pontis, lines and planes exist in this geometry. Thanks.

Explanation / Answer

The following statements on lines, points and planes are evident truth.

Axioms on points and lines:

1. There are at least two distinct points. Here distinct means two points that are considered are different.

2. There is one and only one line that contains two distinct points. This axiom says that given any two distinct points we can draw one and only one line passing through those two points.

3. Every line contains at least two distinct points. The third axiom given by you says every line has exactly two points. But it is a fact that a line is made up of many points. Hence the third axiom should mean any line considered should have two distinct points.

Axioms on points and planes:

4. There are three points that do not all lie on the same line. This means For every line say L made of two distinct points (from the axiom above) there exists a point say P that is not on L which is the third point. If all the three points lie on the same line it is a simple line. This is the basic of a plane.

5. For any three points that do not lie on the same line there is a one and only one plane that contains them. From the above axiom it is clear that all the three points do not lie on the same line. This means a plane is necessary to have all the three points. A line and a point distinct from it. Given any three distinct non-collinear points there exists a single uique plane A that contains them.

6. There are four points that do not all lie on the same plane. This means For every plane say A made of a line and a point (from the axiom above) there exists a point say P that is not on A which is the fourth point. For every plane A there exists a point P that is not on A.

Hope the above explanations are clear. Also there are uncountable points, lines, and planes in geometry.

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