Date: Jan 3, 2013 4:31 AM
Subject: Re: The Distinguishability argument of the Reals.

On Jan 3, 6:26 am, Ralf Bader <> wrote:
> WM wrote:
> > On 1 Jan., 19:19, Zuhair <> wrote:
> >> The distinguishability argument is a deep intuitive argument about the
> >> question of Countability of the reals. It is an argument of mine, it
> >> claims that the truth is that the reals are countable. However it
> >> doesn't claim that this truth can be put in a formal proof.

> > The distinguishability argument is neither deep nor intuitive.
> It is not even an argument, just question-begging.

Call it what may you, what is there is:

(1) ALL reals are distinguishable on finite basis

(2) Distinguishability on finite basis is COUNTABLE.

So we conclude that:

"The number of all reals distinguishable on finite basis must be

Since ALL reals are distinguishable on finite basis, then:

"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.


> > And is
> > not an argument of yours since you do not even understand its
> > implications. It is simply the basis of the axiom of extensionality.
> > How should we distinguish elements if they could not be distinguished?

> > we arrive finally at:
> >> The number of all reals is COUNTABLE.
> > Of course this would be the result if "countable" was a sensible
> > notion.

> You even know what the result would be if non-sensible notions involved
> were sensible. Mückenheim, you are either the Greatest Genius Of All Times
> or one of the greatest idiots.