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)
> > > So we conclude that: > > > "The number of all reals distinguishable on finite basis must be > > countable". > > > Since ALL reals are distinguishable on finite basis, then: > > You seem to use "distinguishable" in two different ways. > > Seeing your argument reminds me of the old chestnut about cats: A cat > has three tails. Proof: No cat has two tails. A cat has one tail more > than no cat. QED. > > There is nothing of that. The intuitive argument of mine here is clear as far as its presentation is concerned, I didn't mention the definition of "distinguishability on finite basis" because it is well known (actually I was asked to SHUT UP by one of the posters because I mentioned explicitly the definition of it?) and because this topic actually comes as a continuation to earlier threads on this topic presented by myself to this Usenet only recently.
Zuhair > > > > > > > "The number of all reals is countable". > > > Because generally speaking no set contain more elements than what it > > CAN have. So you cannot distinguish more reals than what you CAN > > distinguish. Since all reals are distinguished by finite initial > > segments of them, and since we only have COUNTABLY many such finite > > initial segments, then for the first glance it seems that there ought > > to be COUNTABLY many reals so distinguished. This is what our > > intuition would expect! > > > Nobody can say that this simple and even trivial line of thought have > > no intuitive appeal. Definitely there is some argument there, at least > > at intuitive level. > > > However Cantor's arguments all of which are demonstrated by explicit > > and rigorous formal proofs have refuted the above-mentioned intuitive > > gesture, however that doesn't make out of Cantor's argument an > > intuitive one, no, Cantor's argument remains COUNTER-INTUITIVE, it had > > demonstrated a result that came to the opposite of our preliminary > > intuitive expectation. > > > Zuhair