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An IBL Introduction to Geometries

Mark A. Fitch

Section 1.4 Explain and Apply

Checkpoint 1.4.1.

Explain what completeness is and why it is important. Note, an explanation is not a proof, just a clarification for someone else.
Solution.
In a complete system, every statement that is true is provable. Gödel demonstrated that mathematical systems can be either complete or consistent, both or neither.

Checkpoint 1.4.2.

Explain what consistency is and why it is important.
Solution.
In a consistent system, every provable statement is true. Gödel demonstrated that mathematical systems can be either complete or consistent, both or neither.

Checkpoint 1.4.3.

Modify the axioms for the Fano geometry so they are complete and consistent. You might use the Four Point Geometry axioms as a model.
Solution.
The following amended axioms of the Fano geometry are complete and consistent and describe a seven-point Fano geometry:
  1. There exist exactly seven points.
  2. There are exactly three points on every line.
  3. Through every distinct pair of points there is exactly one line.

Checkpoint 1.4.4.

Try to construct a Fano geometry with exactly seven points.
Solution.
The following figure shows a Fano geometry on 7 points.
A diagram of a Fano geometry on 7 points.

Checkpoint 1.4.5.

Modify the axioms of the Five Point Geometry to be complete and consistent. You might use the Four Point Geometry axioms as a model.
Solution.
We will start with making the Five Point Geometry consistent. In attempting to satisfy the axioms, we quickly encounter a contradiction. If we construct a geometry that satisfies Axioms 1, 3, and 4, we find that we are unable to satisfy Axiom 2. We can modify Axiom 2 so that it will be able to be satisfied, but using the Four Point Geometry as a base, we see that the most intuitive modification is to remove Axiom 2. This will also make the modified Five Point Geometry complete, as adding or removing any lines or points will violate an axiom.

Checkpoint 1.4.6.

How many lines are there in this geometry?

Checkpoint 1.4.7.

Modify the axioms of the Five Point Geometry to be complete and consistent and so that they result in five lines.
Solution.
To result in having a complete and consistent Five Point Geometry with 5 lines, we need to modify the fourth axiom so that each line is on exactly 3 points. This will result in our axioms being: (1) There exist exactly five points, (2) There exist exactly five lines, (3) Any two distinct points have exactly one line on both of them, and (4) Each line is on exactly 3 points.
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