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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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 Virgil Posts: 8,833 Registered: 1/6/11
Re: FAILURE OF THE DISTINGUISHABILITY ARGUMENT. THE TRIUMPH OF CANTOR: THE REALS ARE UNCOUNTABLE!
Posted: Jan 9, 2013 3:45 PM

In article
WM <mueckenh@rz.fh-augsburg.de> wrote:

> 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.

Not to those without a language, but the sequence of colors remains for
all who can distinguish each of the colors from the others.
>
> > 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.

Not until you have defined what each digit represents and what digit
string represent, so you are requiring an entire language before getting
a single 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?

Nice try yourself, That is not what he said.

How does one distinguish any one number from a set of infinitely many
numbers of which it is member?

That depends on how the members of that set are accessed. NOte that we
names, such as numerals, or descriptions of what they represent, like
the ratio of lengths of circumference to diameter of a circle.

>
> > 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!

Only the ones we can list can be known to be 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?

But there are always more subsets of a set than members of that set, so
the set of all subsets of a countably infinite set like N will be of
greater cardinality than N itself.
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