1.2: Completeness
- Page ID
- 89843
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Use the following axioms and definitions of intersection and parallel as a definition of the Fano geometry.
more stuff
- There exists at least one line.
- There are exactly three points on every line.
- Not all points are on the same line.
Explore the Fano geometry as follows.
- Draw a line using Geogebra.
- Add a third point to the line.
- Note that Axiom 3 requires one more point. Draw one.
- Must any more lines be added? If so, do so.
- How many lines are in this geometry?
- Add the axiom: every point is on at least one line.
- To your line with three points, and one point not on that line add any lines required by this new axiom.
- Make sure these lines satisfy Axiom 2.
- How many lines are in this geometry?
- May any more lines be added? If so, do so, and be sure the axioms are satisfied.
- How many lines are in this geometry?
- What do the answers to 5, 9, and 11 say about this attempt at constructing a geometry?
- What is needed to fix the difficulty noted in the previous question?