Date: Nov 22, 2012 1:22 PM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 154: Consistency Proof!

On 22 Nov., 17:16, William Hughes <wpihug...@gmail.com> wrote:
> On Nov 22, 12:06 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
>
>
>
>
> > On 22 Nov., 16:27, William Hughes <wpihug...@gmail.com> wrote:
>
> > > On Nov 22, 3:30 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > Can we estimate by means of set theory how many digits *left* to the
> > > > decimal point will be present in the limit (as calculated by
> > > > analytical means) of the real sequence
>
> > > > > > 01.
> > > > > > 0.1
> > > > > > 010.1
> > > > > > 01.01
> > > > > > 0101.01
> > > > > > 010.101
> > > > > > 01010.101
> > > > > > 0101.0101
> > > > > > ...
>
> > > > ?
> > > Yes, The set of digits left of the decimal point is the
> > > empty set.
>
> > This is in contradiction to analysis (although analysis is said to be
> > based upon set theory). Just my point.
>
> > >  Simplest argument.  Start with
>
> > > 100.000...
> > > 10.000...
> > > 1.000...
> > > 0.1000...
> > > 0.01000...
> > > ...
>
> > > The 1 does not exist in the limit.  This 1 corresponds to
> > > the digit with index 5.  We conclude that for
> > > each index the digit corresponding to the digit does
> > > not exist in the limit.  Thus the set of digits in the limit
> > > is the empty set.  Thus, in the limit, the set of digits to
> > > the left of the decimal point is the empty set.
>
> > What has this problem to do with my question?
>
> It answers it.
>
> <I explicitly used
>
> > alternating sequences 010101...
>
> And I deal with the simpler case first

No, yours is a much more difficult case. That's why I conceived the
simplest case.

Regards, WM