Date: Jan 14, 2013 1:40 PM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Finitely definable reals.
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

>

> I've been thinking about this assertion of your and beg to differ.

> 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