Mathematics and Science

[Kuroneko]

He meant any point which is on CF and not in H must be in J. That's perfectly justifiable. The problem is that CF isn't parallel as I defined it.

You're right; I should have caught that when you last qualified your statement. This is indeed the case under your definition, though of course not others (e.g., parallel [lines]: non-intersecting in the same plane). That was pure inattention on my part.

We do have more appropriate terms, such as 'geodesic'

Not quite. Geodesics are beasts of metric geometry, with sometimes quite contradictory properties (e.g., it is not the case that every two distinct points determine one and only one geodesic). I suppose we could work with metric geometry only, and simply specify some constrains (e.g., constant curvature) to regain this property, but there is no requirement of actually doing so. More traditional geometrical axioms work well enough, and have the advantage of not requiring a very substantial amount of background theory to be accessible.

we can take G or ¬G as an axiom and see what comes out. Those would be new axioms.

Of course. Alternatively, we can also do nothing.

If G is the Gödel sentence for a given system, is there any reason to prefer G over ¬G?

no, and that's my point.

[/quote]
If that's the case, then I'm afraid I fail to see the relevance of your comment. If we choose among {G,¬G} to make a new axiom and append it to our system, it will simply be another system. This is fully compatible with my interpretation of axiomatic systems.

[Wyrm]

But CF is not asymptotic to AB, and thus is not a parallel line as I defined it.

Yes, that does change everything, now that I see that you have defined "parallel" in this way to exclude skew lines. Two dimensional thinking strikes again.

However, there are other ways to define "parallel" to exclude skew lines that give you an infinite number of parallel lines satisfying the "at least two" part of your particular axiom. The Euclidean axioms don't tell you what lines, points, ect. are, just how they behave.

[drachefly]

Not quite. Geodesics are beasts of metric geometry, with sometimes quite contradictory properties

interesting... of course the nonuniqueness of spatial geodesics connecting two points is obvious when you think of the existence of multiple images from gravitational lensing.

If that's the case, then I'm afraid I fail to see the relevance of your comment. If we choose among {G,¬G} to make a new axiom and append it to our system, it will simply be another system. This is fully compatible with my interpretation of axiomatic systems.

Well, once you clarified by what you meant by 'axioms define their referents' so it was clear you didn't mean there was a 1-1 correspondence between definitions and axioms, it wasn't so relevant. I was just trying to find axioms that would not have any associated definitions. Either G or ¬G would be such an axiom.

[drachefly]

btw though it's pretty much irrelevant, I wish to make an erratum: My textbook was by Bonola, not Bolyai (one of the inventors of noneuclidean geometry).