```Date: Jan 3, 2013 7:58 AM
Author: Zaljohar@gmail.com
Subject: Re: The Distinguishability argument of the Reals.

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, andthis 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_thdigitof 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 clearas far as its presentation is concerned, I didn't mention thedefinition of "distinguishability on finite basis" because it is wellknown (actually I was asked to SHUT UP by one of the posters because Imentioned explicitly the definition of it?) and because this topicactually comes as a continuation to earlier threads on this topicpresented 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
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