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Topic: Uncountability of the Real Numbers Without Decimals
Replies: 110   Last Post: Dec 10, 2013 4:33 AM

 Messages: [ Previous | Next ]
 Virgil Posts: 8,833 Registered: 1/6/11
Re: Uncountability of the Real Numbers
Posted: Dec 2, 2013 7:59 PM

WM <wolfgang.mueckenheim@hs-augsburg.de> wrote:

> Am Montag, 2. Dezember 2013 21:34:25 UTC+1 schrieb Zeit Geist:
>

> > > > The proof proceeds by Contradiction. We assume the Set of Real Numbers
> > > > is Countable, and thus can be exhausted in a Sequence.

> > > And that is the point at which further reading is absolutely useless.

Only when WM is the one writing.

> > Then why did you keep reading?
>
> I did not. I know Cantor's first "proof" very well.

Not well enough to counter it.

> > > M_1 = empty set, M_{n+1} = (M_n U (n-1, n]) - {q_n}
> > > wher (n, n+1] is the semi-opem interval that contains all rational
> > > numbers q with n < q =< n+1 and q_n is the n-th rational in Cantor's
> > > enumeration of all rationals.

Why does it have to be Cantor's ennumeration of rationals.
All sorts of people have ennumerated the rationals.
I have done it myself several different ways.

> > > The notion of countability requires that lim M_n = { } for n --> oo.
> > > No sober mind will accept that.

Anyone who has seen any ennumeration of the rationals, of which there
are many, should accept that.

And unless WM can name a positive rational that is not one of his own
q_n, he has to accept that eventually every rationsl is excluded.

>
> Your proof is only an implication, a frame. It shows you: If you apply a
> definable sequence, you will get a defined limit. Undefinable sequences will
> not produce anything. No mathematical operation will ever produce an
> undefinable number. Therefore it is useless to think they exist. They do not
> help to resolve Cantor's antinomy that his "proof" of uncountability is done
> by producing defined reals.

Both of Cantor's proofs show that that no attempt to list or count all
real numbers can ever succeed in including all of them. Ther is no
requirement in either that the ones that are shown to be not
counted/listed be undefineable, only uncounted.

> >
> >
> >
> > And please discuss "definable reals" until you have solid Mathematical
> > conception of the idea.
> >

> That is not required for my purpose. It is completely sufficient that one
> finite word will at most define one number. It is not necessary to go into
> the details, since the superset of all finite definitions is countable -
> nothwithstanding how finite definitions are defined.

But the set of real numbers, at least when is complete enough to have
the LUB property cannot be counted.

> But I understand that you have run out of solid arguments.

One of them is quite enough, since WM does not have any.

WE note that WM has never successfully refuted any of the many opposing
solid arguments that others have presented here.
--

Date Subject Author
12/2/13 Tucsondrew@me.com
12/2/13 William Elliot
12/2/13 Tucsondrew@me.com
12/2/13 G. A. Edgar
12/2/13 Tucsondrew@me.com
12/2/13 wolfgang.mueckenheim@hs-augsburg.de
12/2/13 gnasher729
12/2/13 wolfgang.mueckenheim@hs-augsburg.de
12/2/13 Virgil
12/2/13 Tucsondrew@me.com
12/2/13 Virgil
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 Tucsondrew@me.com
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Virgil
12/3/13 Virgil
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Tucsondrew@me.com
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 Virgil
12/5/13 Virgil
12/10/13 Robin Chapman
12/2/13 Virgil
12/2/13 wolfgang.mueckenheim@hs-augsburg.de
12/2/13 Virgil
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 Virgil
12/2/13 Tucsondrew@me.com
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 Tucsondrew@me.com
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Tucsondrew@me.com
12/5/13 Michael F. Stemper
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 Virgil
12/6/13 wolfgang.mueckenheim@hs-augsburg.de
12/6/13 Tucsondrew@me.com
12/6/13 wolfgang.mueckenheim@hs-augsburg.de
12/6/13 Virgil
12/6/13 Brian Q. Hutchings
12/7/13 Brian Q. Hutchings
12/7/13 Brian Q. Hutchings
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 fom
12/7/13 albrecht
12/7/13 fom
12/7/13 ross.finlayson@gmail.com
12/8/13 albrecht
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 fom
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 fom
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 Virgil
12/7/13 fom
12/8/13 Virgil
12/7/13 Virgil
12/7/13 Virgil
12/7/13 Virgil
12/8/13 albrecht
12/6/13 Virgil
12/6/13 Virgil
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 Virgil
12/5/13 Virgil
12/3/13 Virgil
12/3/13 Michael F. Stemper
12/3/13 Virgil
12/3/13 fom
12/2/13 Tucsondrew@me.com
12/2/13 wolfgang.mueckenheim@hs-augsburg.de
12/2/13 Virgil
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 Virgil
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Virgil
12/2/13 Virgil
12/2/13 Tucsondrew@me.com
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 Virgil
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 gnasher729
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 gnasher729
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Virgil
12/3/13 Virgil
12/3/13 Virgil
12/5/13 gnasher729
12/3/13 Tucsondrew@me.com
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Tucsondrew@me.com
12/5/13 Tucsondrew@me.com
12/5/13 Virgil
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 Tucsondrew@me.com
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Tucsondrew@me.com
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 Virgil
12/8/13 wolfgang.mueckenheim@hs-augsburg.de
12/8/13 Virgil
12/5/13 Virgil
12/3/13 Virgil
12/2/13 ross.finlayson@gmail.com
12/4/13 ross.finlayson@gmail.com
12/3/13 albrecht
12/3/13 Tucsondrew@me.com
12/5/13 albrecht
12/5/13 Tucsondrew@me.com