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Discuss if it is a model of geometry, by verifying the 3 axioms one by one.

Discuss if it is a model of geometry, by verifying the 3 axioms one by one.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Verify by showing that they follow the three incidence axioms:

(I1) Given any two discrete points there exists a unique line containing them
(I2) Given any line there exists at least two distinct points lying on it
(I3) There exists three non-collinear points

We will use Hilbert’s axiom system that is constructed with the three primitive terms point, line and plane (lie on).

¹A great circle is the intersection of $2 with any plane through the origin of R³.
Transcribed Image Text:¹A great circle is the intersection of $2 with any plane through the origin of R³.
1. S² Consider the S² Geometry, whose interpretation is the following: • point: Point on the surface of the unit sphere s² = {(x, y, z) = R³|x² + y² + z² = 1}. line: The great circles¹ of the unit sphere. ● lie on: A point p (x, y, z) = S² lies on a great circle l if p = l. Discuss if it is a model of geometry, by verifying the 3 axioms one by one. =
Transcribed Image Text:1. S² Consider the S² Geometry, whose interpretation is the following: • point: Point on the surface of the unit sphere s² = {(x, y, z) = R³|x² + y² + z² = 1}. line: The great circles¹ of the unit sphere. ● lie on: A point p (x, y, z) = S² lies on a great circle l if p = l. Discuss if it is a model of geometry, by verifying the 3 axioms one by one. =
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