Date: Feb 4, 2013 3:21 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 203
On 3 Feb., 23:09, William Hughes <wpihug...@gmail.com> wrote:

> On Feb 3, 10:58 pm, WM <mueck...@rz.fh-augsburg.de> wrote:

>

>

>

>

>

> > On 3 Feb., 22:29, William Hughes <wpihug...@gmail.com> wrote:

>

> > > > > We can say "every line has the property that it

> > > > > does not contain every initial segment of s"

> > > > > There is no need to use the concept "all".

>

> > > > Yes, and this is the only sensible way to treat infinity.

>

> > > So now we have a way of saying

>

> > > s is not a line of L

>

> > > e.g. 0.111... is not a line of

>

> > > 0.1000...

> > > 0.11000...

> > > 0.111000....

> > > ...

>

> > > because every line, l(n), has the property that

> > > l(n) does not contain every initial

> > > segment of 0.111...

>

> > But that does not exclude s from being in the list.

>

> It certainly excludes 0.111... from being a single line

> of the list.

>

> So 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

> ?-

Of course, every FIS is in a line. And every FIS contains all

preceding FIS.

This list

0.1

0.11

0.111

...

contains every FIS of 0.111... as well in the (unchanged) diagonal as

in a line. Obviously the diagonal cannot have more 1' than every line.

(Of course you cannot expect to define 0.111... without a last element

1_omega but to find a last line g.)

Regards, WM