Given a hypothetical process that selects real numbers entirely at random, the numbers thus selected are always of the "undeterminable" type. Essentially, think of a number with an infinite and entirely random sequence of decimal digits. Determinable numbers can have an infinite sequence of decimal digits, but the digits are never entirely random, since they are constrained by the function that creates the number.
I don't know if this is proof that this set of undeterminable numbers is infinitely larger than the set of determinable numbers (thus the apparent probability of selecting a determinable number being zero) or that this proof relies on the fact that the set of undeterminable numbers is infinitely larger than the set of determinable numbers. The argument seems to be headed towards an actual proof that the undeterminable set must be infinitely larger.
Regarding the case of a number never being selected twice - Even though a number was selected, and cannot be selected again (P = 0), it is still an indeterminable number and we can never really know what that number was (because of the infinite series of random decimal digits).
Note: While undeterminable numbers are a subset of transcendental numbers (because all non transcendental numbers are determinable), all transcendental numbers are not undeterminable.
On Nov 26, 2013, at 2:46 PM, Robert Hansen <firstname.lastname@example.org> wrote:
> > On Nov 26, 2013, at 1:38 PM, Joe Niederberger <email@example.com> wrote: > >> I think if you model a light ray as a 1 dimensional line, >> that its just an approximation good for some purposes and not for others. Likewise, number lines and points and infinitely sharp darts makes a nice mental image, but the absurd result that says the dart does hit a particular point, but that it has zero % chance of hitting any point in particular, tells you something is not quite right with that picture. Probability is an area of mathematics where the difference between models, physics, and math is often murky, though I don't think it has to be. > > We are using the ray of light more like a metaphor. In the end, all we are doing is selecting a number by an entirely random and natural process. In that context, I think the use of probability is valid. That doesn't mean the execution of the argument is valid though. I am still thinking that through. > > Bob Hansen > >