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