Date: Jan 3, 2013 7:58 AM
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, 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
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.
> > "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