Search All of the Math Forum:

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

Topic: Finitely definable reals.
Replies: 52   Last Post: Jan 18, 2013 2:37 PM

 Messages: [ Previous | Next ]
 mueckenh@rz.fh-augsburg.de Posts: 18,076 Registered: 1/29/05
Re: Finitely definable reals.
Posted: Jan 14, 2013 1:40 PM

On 14 Jan., 14:23, forbisga...@gmail.com wrote:
> On Saturday, January 12, 2013 12:24:46 AM UTC-8, WM wrote:
> > On 12 Jan., 02:14, Virgil <vir...@ligriv.com> wrote:
>
> > > > Cantor managed to prove that there are more than countably many finite
>
> > > > binary strings possible. Remember, the part behind a_nn of a_n is not
>
> > > > relevant for his proof.
>
> > > Quite so, but that in no way weakens his proof.
>
> > It shows a self-contradiction by the fact that there must be an
>
> > antidiagonal that from every entry differs at a finite place. But if
>
> > the list is complete with respect to all finite binary strings, this
>
> > is obviously impossible.
>
> > Regards, WM
>
> The decimal expansion of 1/3 only differs from all other reals at
> the infinite.

What do you understand by this statement?

>  It takes the infinite to make it 1/3.  When one multiplies
> a number by 10 one moves the decimal place one position to the right.
> Only the infinite decimal expansion will do to restore the fraction.
> Any finite expansion will have a delta from 1/3.

In my opinion everything that in mathematics can be used to express
1/3 as a decimal fraction is "all its finite digits". That means, only
in the infinite we obtain 1/3, but a better phrase describing "the
infinite" is simply "never". We will never obtain 1/3 as a decimal.
Alas, if Cantor was right, we must assume that "never" is a certain
point in time that can be reached and surpassed. Therefore Cantor was
very happy when he read in the Holy Bible "Dominus regnabit in eternam
*et ultra*" (emphasis by Cantor).

Regards, WM