Geodesic Curves Covering Surfaces

In summary: A geodesic will intersect a coordinate patch in countably many segments simply because any open subset of the real line has at least countably many points in it.
  • #36
Hurkyl said:
The version on Wikipedia doesn't put a limitation on how many countable subsets you use.

Anyways, I don't think validity is the question you wanted to ask; it's not a theorem of ZFC, it's invalid in any model of ZFC+CH, and valid in any model of ZFC+AX.

I was content with maze's response -- I asked, and he answered, I didn't feel it important to press on. But my main reaction is simply that the criterion seems esoteric; it doesn't appear to obviously boil down to anything that I can imagine people having strong opinions about.

If AX were accepted, it would be a refutation of CH. People might have strong opinions about that. Should AX be rejected because it's 'esoteric'?
 
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  • #37
I never said it should. Being esoteric just makes it hard to have opinions about. :smile:

The only times I've ever really ran into the CH are:
1. It let's you use [itex]\aleph_1[/itex] to refer to |R|
2. It simplifies the classification of real closed fields

Point (1) is highly superficial, and I don't work with real closed fields enough to have any string opinions about point (2). While AX is also related to CH, I have much less connection to it than these other two points.
 
<h2>1. What are geodesic curves?</h2><p>Geodesic curves are the shortest paths connecting two points on a surface. They are the equivalent of straight lines in Euclidean geometry, but instead follow the curvature of the surface.</p><h2>2. How do geodesic curves cover surfaces?</h2><p>Geodesic curves cover surfaces by connecting every point on the surface to form a continuous network of curves. This network of curves provides a way to navigate and measure distances on the surface.</p><h2>3. What is the significance of geodesic curves?</h2><p>Geodesic curves have several important applications in mathematics, physics, and engineering. They are used in the field of differential geometry to study the properties of curved surfaces and in the field of geodesy to measure distances on the Earth's surface. They also have practical applications in navigation and mapping.</p><h2>4. How are geodesic curves calculated?</h2><p>The calculation of geodesic curves depends on the specific surface being studied. For simple surfaces, such as spheres or cylinders, the equations for geodesic curves can be derived analytically. For more complex surfaces, numerical methods are used to approximate the geodesic curves.</p><h2>5. Can geodesic curves exist on non-curved surfaces?</h2><p>No, geodesic curves can only exist on curved surfaces. On flat surfaces, such as a plane, the shortest path between two points is a straight line. Geodesic curves are a result of the curvature of the surface and do not exist on flat surfaces.</p>

1. What are geodesic curves?

Geodesic curves are the shortest paths connecting two points on a surface. They are the equivalent of straight lines in Euclidean geometry, but instead follow the curvature of the surface.

2. How do geodesic curves cover surfaces?

Geodesic curves cover surfaces by connecting every point on the surface to form a continuous network of curves. This network of curves provides a way to navigate and measure distances on the surface.

3. What is the significance of geodesic curves?

Geodesic curves have several important applications in mathematics, physics, and engineering. They are used in the field of differential geometry to study the properties of curved surfaces and in the field of geodesy to measure distances on the Earth's surface. They also have practical applications in navigation and mapping.

4. How are geodesic curves calculated?

The calculation of geodesic curves depends on the specific surface being studied. For simple surfaces, such as spheres or cylinders, the equations for geodesic curves can be derived analytically. For more complex surfaces, numerical methods are used to approximate the geodesic curves.

5. Can geodesic curves exist on non-curved surfaces?

No, geodesic curves can only exist on curved surfaces. On flat surfaces, such as a plane, the shortest path between two points is a straight line. Geodesic curves are a result of the curvature of the surface and do not exist on flat surfaces.

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