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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 5:41 PM

In article
WM <mueckenh@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.
>
> > 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.
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