Maybe it’s a hijack, but I find the negation of the Axiom of Choice much more intuitive than the Axiom of Choice itself.
Mathematician: And now, I can choose one member of each of these sets.
Me: OK, then do it. Choose one member of each set.
Mathematician: Done!
Me: So, which member did you choose for each of them?
Mathematician: Oh, I don’t know which one I chose. But I can do it.
Me: But if you don’t know which one you chose, what do you mean that you chose them?
I thought about that before I saw your post, though these are not quite rays in the sense that other posters are describing, since v ~ -v. Better to picture them as lines through the origin, or, to a geometer, elements of the projective space (at least if the origin is deleted first).
I’m sure the OP’s “anti-rays” show up in some construction somewhere; projective spaces commonly occur everywhere (not that that answers his question).
How about, consider an attracting fixed point of a dynamical system. Then orbits of points in the surrounding basin of attraction will be “anti-rays” that converge to the point.
Maybe I’m misunderstanding something, but isn’t your anti-ray exactly the same object as a ray? Just from a different perspective?
What does this have to do with anything?
Yes, you have. That’s not how “paradox” is defined in any context I know of.
Well, a ray…radiates…from an origin. Is it different if the “end” is a destination? An object which absorbs rather than emits?
There are different ways to think of it.
One way is that a ray (or a line, or a line segment, or whatever) is just a set of points that are all just considered as a whole, and there’s no sense of action or motion (radiating, absorbing, emitting) involved.
Another way is that, to someone like Euclid, there is no “completed infinity,” so you can’t get a ray that ends at a definite point because there’s no “point at infinity” to start from.
And your way is yet another way, incompatible with those above, but not therefore “wrong.”
See Post #23-- apparently you are thinking of dynamics, say of a point moving in the plane as a function of time, in which case there is an obvious difference between a trajectory that starts at or near a point and shoots off somewhere, and one that does the opposite.
Here is a specific example I stole from Wikipedia: look at the leftmost figure in https://upload.wikimedia.org/wikipedia/en/5/55/LinearFields.png
And yet there is such a thing as a long line and long ray. They are also infinitely long. But at the same time, they are longer than the “normal” infinite line and ray.
(roughly speaking, the long line is composed of uncountably many unit-length segments, whereas a normal line has only countably many segments)
