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What is the average number of intersections for two infinite curves confined to a plane?
How long is a piece of string?What is the average number of intersections for two infinite curves confined to a plane?
Is the limit of a random walk as the step size goes to zero well-defined?How do you define a random curve? A random walk is discrete, but random curve?
Choosing an arbitrary element from a set is a random process.No, when you just "choose" an element out of the set of all curves it isn't "random" in the probabilistic sense. First of all can we even define a probability measure on that set? How big is it? It might be bigger than the set of reals.
Two infinite curves may have 0 to infinity intersections. The average of any (finite or infinite) set of numbers that are non-negative and contains at least 1 infinity is equal to infinity. Thus the answer to the original question is, statistically and definitively, infinity.It may turn out that infinite sets cannot be averaged or proportioned, but if they can, I believe one of your answers is the correct number.
There are beth_2 functions from R to R, so their set is larger than the set of reals. But the set of smooth curves from R to R is only as large as the set of reals (beth_1).No, when you just "choose" an element out of the set of all curves it isn't "random" in the probabilistic sense. First of all can we even define a probability measure on that set? How big is it? It might be bigger than the set of reals.
True, it might be a fractal. But I think a reasonable definition of intersection would be to consider curves to intersect if for some epsilon > 0 the random walks of step size epsilon intersect. (I can make this precise if desired, but there's nothing exciting about it.)The problem with the step size approach is that the limit might not be a curve, so there's no sense of talking about "intersecting" itself.
Yes, |R^R| is bigger than |R|. I'm pretty sure the curves on R^2 are upper-bounded by |R^R| = beth_2 -- just define them parametrically.The set of all curves in R^2 is probably larger than the set of all reals. And the upper bound is the size of the set of all functions from R to R.
Nah, I don't think that's reasonable. I'd much sooner disregard sets of measure 0 than have them override the rest of the set if it came to that.The average of any (finite or infinite) set of numbers that are non-negative and contains at least 1 infinity is equal to infinity.