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Re: The Distinguishability argument of the Reals.
Posted:
Jan 3, 2013 7:23 AM
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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.
> 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.
> "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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