
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 UTC8, 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 selfcontradiction 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

