On 03/01/2013 8:58 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)
Exactly. This is precisely what I wrote. IF you have TWO *DIFFERENT* reals r1 and r2, then you can establish this fact in finite time. However, if you are given two different descriptions of the *SAME* real, you will have problems. How do you find out that NOT exist n... in finite time?
Moreover, being able to distinguish two reals at a time has nothing at all to do with the question of how many there are, or how to distinguish more than two. Your (2) uses a _different_ concept of distinguishability.