Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

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

 Messages: [ Previous | Next ]
 wolfgang.mueckenheim@hs-augsburg.de Posts: 2,004 Registered: 10/18/08
Re: Uncountability of the Real Numbers Without Decimals
Posted: Dec 2, 2013 4:23 PM

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.
>
>
> Then why did you keep reading?

I did not. I know Cantor's first "proof" very well.
>
>
>

> > 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.
>
> >
>
>
>
> So, an enigmatic s of the rationals is possible?

No. But if you think that it is possible, you can see it being contradicted here.

>
> > The notion of countability requires that lim M_n = { } for n --> oo.
>
>

> > No sober mind will accept that.
>
>
> How do you know q_n is in your semi-open interval and isn't added in later?

I start with the enumeration: 1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, ... The numbers q_n are always in the intervals already added.
>
> > > I find the proof rather straight forward. Question, comments, suggestions and corrections are welcome.
>
>

> > What step produces the first undefinable number from a given set of defined numbers?

>
> None or maybe any. The proof just uses existence non-empty intervals. Unless you think some of those intervals are empty?

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. (Do not think that you get an undefinable limit if you refrain from defining a sequence.)
>
>
>
> 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.

Look here: It is mathematical fact that all rationals that are less than 1 are less than 2. For that sake I need not answer the question what the largest rational less than 1 could be.

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

Regards, WM

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