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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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 mueckenh@rz.fh-augsburg.de Posts: 18,076 Registered: 1/29/05
Re: FAILURE OF THE DISTINGUISHABILITY ARGUMENT. THE TRIUMPH OF
CANTOR: THE REALS ARE UNCOUNTABLE!

Posted: Jan 9, 2013 12:15 PM

On 9 Jan., 13:24, Zuhair <zaljo...@gmail.com> wrote:

> That was the distinguishability argument, presented by an intuitive
> simile.

Your colors correspond to characters. A finite sequence of colors
corresponds to a word.

> To describe this color change we us TRIPLETS so we write:
>
> {{G,R,R}, {G,Y,Y}, {G,G,Y}, {R,R,R}, {Y,Y,Y}, {G,G,G}}
>
> So shirt changing did result in discriminating ALL six members of the
> group.
>
> This had been achieved by recognizing the pattern of change of color
> of shirts upon re-wearing of them.

Recognizing the change of digits of a string you get a number.

> When we distinguish a real r from all other reals we are not doing it
> with ONE initial finite segment of r, no we are doing it with actually
> countably infinite many finite initial segments of r,

Nice try! You want to prove finite distinguishability by an infinite
set? When have you finished your last definition of a real? If you
think it over a little bit, then you will agree: We, you and me and
everybody else, define a real with one finite word like "real solution
of x^3 = 2" or "crt(2)".
Have you ever tried to define crt(2) by its infinite sequence of
digits?

> Actually Cantor's diagonal argument
> proves that the number of those tuples MUST be uncountable.

In fact, it does! But we know that the number of those tuples is
countable!

> No problem
> since even at intuitive level distinguishability by shift recognition
> is not necessarily limited to the total number of distinguishing
> objects as demonstrated in the example above, it is related to the
> number of tuples of the distinguishing objects, and of course those
> can indeed be more than the number of distinguishing objects!

Even if your clumsy example was acceptable, you could not leave the
realm of countability, because finite times finite is finite and even
countable times countable is countable. Or would you like to see
uncountably many changes of shirts?

> All of WM's kind of speech presented to this Usenet belong to the last
> form of misleading guided intuitions, and this argument is the main
> one that underneath his argument, and it is shown here clearly to be
> FALSE, it is a Pseudo-intuitive argument.

Do you know the axiom of extensionality? Do you think that in a formal
theory undistinguishable elements or sets containing undistinguishable
elements can be distinguished, formally or with aleph-order logic?

Regards, WM