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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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 JT Posts: 1,434 Registered: 4/7/12
Re: FAILURE OF THE DISTINGUISHABILITY ARGUMENT. THE TRIUMPH OF
CANTOR: THE REALS ARE UNCOUNTABLE!

Posted: Jan 9, 2013 6:04 PM

iOn 9 Jan, 23:41, Virgil <vir...@ligriv.com> wrote:
> In article
>
>  WM <mueck...@rz.fh-augsburg.de> wrote:

> > 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.
>
> No one need do so when not trying to pass one of WM's courses.
>
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>

> > > 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.

>
> Even the best may have lapses, and WM quote mines those lapses with much
> greater effectiveness than he manages to do mathematics.
>
>
>

>
> Since WM only manages to think with someone else's head, if at all, he
> should reck his own rede.
>

> > Construct a list of all rationals of the unit interval. Remove all
> > periodic representations and keep only the finite ones

>
>  Since any sensible method of listing of rationals will not be done in
> terms of their decimal or other base expansions, why do it the hard way?
>
> , 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.

>
> There will be no finite digit position in the list up to which position
> the list itself does not contain duplicates, however any nonterminating
> decimal will not be in any list such as WM has described, so his list is
> incomplete.
>

> > That means it can never deviate
> > from every entry.

>
> On the contrary, if every listed decimal terminates then every
> non-terminating decimal is unlisted.
>

> > According to Cantor, anti-diagonal deviates from
> > every entry of the list at a finite position.

>
> In AT LEAST one finite position!
>
> An antidiagonal might even differ from some entries at EVERY digit
> position.
>

> > But even according to
> > Cantor it deviates never(at no finite position) from all entries.

>
> WM's Quantifier dyslexia strikes again.
>
> In order for an infinite sequence of digits to differ from all the
> infinite sequences of digits in a list, it only needs to differ suitably
> from each entry at one digit position, and that position can be
> different for different entries in the list.
>
> And any such infinite list of infinite sequences of digits, from a set
> of two or  more digits, is provably incomplete since Cantor showed that
> there is always a sequence missing from that list.
> --

I beleive answers is at a very basic level, what distuingish a natural
from a real and so on. I do not know much about math but it seem to me
that this must be its foundation.