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Topic: FAILURE OF THE DISTINGUISHABILITY ARGUMENT. THE TRIUMPH OF CANTOR:
THE REALS ARE UNCOUNTABLE!

Replies: 47   Last Post: Jan 12, 2013 11:33 AM

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 Zaljohar@gmail.com Posts: 2,665 Registered: 6/29/07
Re: FAILURE OF THE DISTINGUISHABILITY ARGUMENT. THE TRIUMPH OF
CANTOR: THE REALS ARE UNCOUNTABLE!

Posted: Jan 10, 2013 12:47 PM

On Jan 10, 1:06 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 10 Jan., 04:52, Zuhair <zaljo...@gmail.com> wrote:
>

> > The reason is that we can have even at finite basis more n sized
> > tuples of m values than n x m.

>
> Of course,, when order plays a role, then you have 10^3 tuples of
> lenght 3 when using 10 colors. Who told you else? But if you use
> length aleph_0, then you leave the finite realm - whether you do it
> with digits or with finite initial segments.
>
> Regards, WM

What finite realm, I'm speaking about Omega_sized tuples of finite
initial segments of reals, so each tuple have countably infinite
number of entries, and each entry range over countably infinite number
of values (because the total number of finite initial segements of
reals is countable) so we would have Aleph_0 ^ Aleph_0 of such tuples,
and not as you thought Aleph_0 x Aleph_0.

What I'm saying is that the distinguishability argument cannot
establish proving countability of the reals from the fact that we have
countably many initial segements of them, because the distinguishing
process of each real resembles intuitively speaking my example number
2 where a change in color of shirts can manage to distinguish more
persons than the number of colors of shirts after which they are
distinguished per each trial of wearing of shirts. So accordingly we
can discriminate a number of reals up to the number of all Omega_sized
tuples of finite initial segments of reals, and this would be Aleph_0
^ Aleph_0 and we have NO intuitive justification to say that the
number of such tuples is countable. That is an argument made by
INTUITIVE analogies using similes that are fairly natural. Anyhow I do
conceded that using such intuitive similes is not that easy to grasp,
many people would find it difficult to follow. However the result is
that there is NO intuitive grounds to say that the number of all reals
are countable.

Zuhair