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