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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 9, 2013 12:57 PM

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!
>

> Regards, WM

No we don't know that! there is NO seal on how many "Omega" sized
tuples of FINITE initial segments of reals do we have! there is no
argument (even at intuitive level) in favor of having countably many
such Omega sized tuples. 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!!! Since we have
Uncountably many Omega sized tuples of finite initial segments of
reals, then we do have UNCOUNTABLY many reals distinguished after
them! that's all fairly intuitive, there is nothing wrong with the
line of that reasoning at all. 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, so I don't see why you keep asking such SILLY
irrelevant questions.

Zuhair