Date: Feb 4, 2013 6:21 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 203

On 4 Feb., 11:19, William Hughes <wpihug...@gmail.com> wrote:
> > Of course, every FIS is in a line.
>
> True but irrelevant.  We can use induction to
> show that there is no natural number n, such
> that the nth line of L contains every FIS
> of 0.111....


We can use induction to show that there is no natural number which
would allow us to draw any final conclusion, i.e., there are
infinitely many lines remaining beyond line number n.
>
> The question is now
>
> Can a potentially infinite list
> of potentially infinite 0/1
> sequences have the property that
> if s is a potentially infinite 0/1
> sequence, then there is a line, g, of L
> with the property that every
> initial segment of s is contained in g
> ?
>
> Yes or No please


No. There cannot be a line g where the FIS are complete, because the
FISs cannot be complete at all. But I already showed you the list
which satisfies the possible claim: Every FIS is in a line.

0.1
0.11
0.111
...

(Because there is no FIS of the diagonal that is missing in every
line.)

By the way, the FISs are isomorphic to the natural numbers. There
cannot be a line g where the natural numbers are complete.

Regards, WM