On 1/3/2013 7:52 AM, gus gassmann wrote: > 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. > >
It sure is nice to see the problem in the definition.