GeoGebra Classroom

# Independence of I1, I2, I3, and P

Prove that the axioms I1, I2, I3 and P are independent of each other. (ie. You cannot prove any one as a result of assuming the others.)

## Axioms of Incidence and P

I1. For any two distinct points, A and B, there exists a unique line containing A and B. I2. Every line contains at least two points. I3. There exists three noncollinear points (that is, three points not all contained in a single line). P. For each points A and each line , there is at most one line containing A that is parallel to .

## Independence of I1

Proof [By Counterexample]: Assume that I1 is dependent on the other Incidence Axioms and Axiom P. Consider two lines, and . According to I2, there are at least two points on each line. Therefore, place points A and B on and C and D on . Since there are four points in this geometry, we can see that I2 is satisfied which allows for at least three noncollinear points. We can also see that P is satisfied because line is parallel to the line which contains point A. However, notice that I1 is not satisfied because there is not a unique line drawn between each of there points in the geometry. Therefore, we can conclude that I1 is independent of the other axioms.

## Independence of I2

Proof [By Counterexample]: Assume that each of the axioms of incidence and P are dependent. Consider the points A, B, and C. I1 gives us unique lines between each of these points. I3 is satisfied because there are three noncollinear points by construction. Also notice that to satisfy I3, there needs to exist two lines and . Notice that by construction these two lines are parallel to one another so that P is satisfied. However, notice that line contains only point C. Therefore, we can conclude that I2 is not satisfied and that I2 is independent of the other axioms.

## Independence of I3

Proof [By Counterexample]: Assume I3 is dependent on the other axioms. Consider lines and which are parallel by construction and the points A,B,C,D and E that lie on those lines. Notice that I1 is satisfied because there are unique lines between each of the points in the geometry. Also notice that each line contains at least two points. By construction, we know that P is satisfied because and are parallel. However, note that there are three collinear points on line which contradicts I3. Therefore, I3 is independent of the other axioms of incidence.

## Independence of P

Proof [By Contradiction]: Assume P is dependent on the other axioms. Consider points A,B, and C and lines and . Notice that each of the lines lie on at least two points and that each set of two points share a unique line which satisfies both I1 and I2. Also, notice that there are three noncollinear points which satisfied I3. However, notice that P is not satisfied because there are no parallel lines. Therefore, P can not be dependent on the other axioms.