Date: Feb 4, 2013 6:33 AM
Author: William Hughes
Subject: Re: Matheology § 203

On Feb 4, 12:21 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> 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....

>
> <snip> [For every n] there are
> infinitely many lines remaining beyond line number n.


This does not prevent us from using induction to show that
there is no natural number n, such
that the nth line of L contains every FIS
of 0.111....
>
>
>

> > 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.



So we have potentially infinite sets like |N
where you can say

If L is a potentially infinite list of
natural numbers then can have the property

If n is a natural number then n is a line of L

and potentially infinite sets like
the potentially infinite 0/1 sequences
where you cannot say

If L is a potentially infinite list
of potentially infinite 0/1 sequences

then if s is a potentially infinite
sequence then s is a line of L