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, 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