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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 4:23 PM

On 9 Jan., 18:57, Zuhair <zaljo...@gmail.com> wrote:
> On Jan 9, 8:15 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>

> > On 9 Jan., 13:24, Zuhair <zaljo...@gmail.com> wrote:
>
> > > 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 we don't know that!

Perhaps you don't. But everybody can learn it.

> there is NO seal on how many "Omega" sized
> tuples of FINITE initial segments of reals do we have!

Omega-sized sequences of finite initial segments are not different
from omega-sized sequences of digits. Both do not belong to the set of
finite definitions. Do you really believe that by using finite initial
segments instead of simple digits you make infinite sequences finite?

> there is no
> argument (even at intuitive level) in favor of having countably many
> such Omega sized tuples.

But there is a simple argument that omega sized tuples are infinite
and are not suitable as finite definitions.

> Actually the ONLY arguments that we have is
> in favor of having uncountably many such Omega sized tuples, which are
> Cantor's arguments. The set of All finite initial segments of reals is
> COUNTABLE! Yes! BUT the set of all Omega sized tuples of finite
> initial segments of reals is NOT countable!!!

That is not of interest. A Cantor-list does not use omega-sized tuples
but only finite initial segments.

> And also I want to stress that all of
> what I'm saying is abiding by the axiom of Extensionality of course,
> no doubt, and accordingly I do NOT hold that in a formal consistent
> theory we can distinguish UNdistinguishable elements or objects, since
> this is clearly nonsense. However the concept of Uncountability
> doesn't lead to that,

Try to read the literature of great mathematicians, for instance
matheology § 187. Or Bernays: If we pursue the thought that each real
number is defined by an arithmetical law, the idea of the totality of
real numbers is no longer indispensable. Or Weyl: Die möglichen
Kombinationen endlichvieler Buchstaben bilden eine abzählbare Menge,
und da jede bestimmte reelle Zahl sich durch endlichviele Worte
definieren lassen muß, kann es nur abzählbar viele reelle Zahlen geben
- im Widerspruch mit Cantors klassischem Theorem und dessen Beweis. Or
Schütte (Hilbert's last student): Definiert man die reellen Zahlen in
einem streng formalen System, in dem nur endliche Herleitungen und
festgelegte Grundzeichen zugelassen werden, so lassen sich diese
reellen Zahlen gewiß abzählen, weil ja die Formeln und die
Herleitungen auf Grund ihrer konstruktiven Erklärungen abzählbar
sind.

Or try to think with your own head:
Construct a list of all rationals of the unit interval. Remove all
periodic representations and keep only the finite ones, but all.
Then the list contains all finite initial segments which a possible
anti-diagonal can have. So the anti-diagonal can at no finite position
deviate from every entry of the list. That means it can never deviate
from every entry. According to Cantor, anti-diagonal deviates from
every entry of the list at a finite position. But even according to
Cantor it deviates never(at no finite position) from all entries.