On 1/3/2013 6:58 AM, Zuhair wrote: > On Jan 3, 3:23 pm, gus gassmann <g...@nospam.com> wrote: >> On 03/01/2013 5:31 AM, Zuhair wrote: >> >>> Call it what may you, what is there is: >>> (1) ALL reals are distinguishable on finite basis >> >> > (2) Distinguishability on finite basis is COUNTABLE. >> >> What does this mean? If you have two _different_ reals r1 and r2, then >> you can establish this fact in finite time. The set of reals that are >> describable by finite strings over a finite character set is countable. >> However, not all reals have that property. > > I already have written the definition of that in another post, and > this post comes in continuation to that post, to reiterate: > > r1 is distinguished from r2 on finite basis <-> > Exist n. n is a natural number & Exist d_n, k_n: d_n is the n_th > digit > of r1 & k_n is the n_th digit of r2 & d_n =/= k_n)
There is a difference between saying ALL REALS are distinguishable and what you have written above. What you have written asserts that EVERY PAIR OF DISTINCT REALS have discernible representations. It simply says that the representation of the real numbers to which it refers can serve as canonical names.
Dedekind cuts define all reals.
Cantor fundamental sequences define all reals.
The Euclidean algorithm does not define any reals.
You may, as WM does, deny uses of a completed infinity.
You may, as Skolem does, question the meaningfulness of nondenumerability with respect to finitely-generated countable languages.
But you need to recognize that the problem here is a poorly constructed statement.