Date: Jan 31, 2013 8:54 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 203

On 31 Jan., 14:41, William Hughes <wpihug...@gmail.com> wrote:
> On Jan 31, 10:23 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
>
>
>
>

> > On 31 Jan., 10:04, William Hughes <wpihug...@gmail.com> wrote:
>
> > > On Jan 31, 9:48 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > > On 30 Jan., 22:38, William Hughes <wpihug...@gmail.com> wrote:
>
> > > > > On Jan 30, 10:24 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > > > > On 30 Jan., 22:14, William Hughes <wpihug...@gmail.com> wrote:
>
> > > > > > > On Jan 30, 6:06 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > > > > > > On 30 Jan., 12:32, William Hughes <wpihug...@gmail.com> wrote:
>
> > > > > > > > > On Jan 30, 12:21 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > > > > > > > > On 30 Jan., 12:02, William Hughes <wpihug...@gmail.com> wrote:
>
> > > > > > > > > > > Summary.  We have agreed that
>
> > > > > > > > > > > For a potentially infinite list L, the
> > > > > > > > > > > antidiagonal of L is not a line of L.

>
> > > > > > > > > Do you agree with the statement
>
> > > > > > > > > For a potentially infinite list, L,
> > > > > > > > > of potentially infinite 0/1 sequences
> > > > > > > > > the antidiagonal of L is not a line
> > > > > > > > > of L

>
> > > > > > > > Yes, of course. We have a collection of which we can keep a general
> > > > > > > > overview. And in finite sets (potential infinity is nothing but finity
> > > > > > > > without an upper threshold) "for every" means the same as "for all".
> > > > > > > > There is no place to hide.

>
> > > > > > > So now we have
>
> > > > > > > For a potentially infinite list, L,
> > > > > > > of potentially infinite 0/1 sequences
> > > > > > > the antidiagonal of L is not a line
> > > > > > > of L

>
> > > > > > > Can a potentially infinite list, L,
> > > > > > > of potentially infinite 0/1 sequences
> > > > > > > have the property that every
> > > > > > > potentially infinite 0/1 sequence
> > > > > > > is a line of L?

>
> > > > > > Potential infinity is the opposite of completeness like "infinite" is
> > > > > > the opposite of "finished". So *every* line number n would not imply
> > > > > > *all* possible line numbers of the set |N defined by AxInf.

>
> > > > > This does not answer the question.  Please answer the question.-
>
> > > > The question is not properly defined.
> > > > Do you mean "every" in the potential sense of "all from 1 to n"? Or do
> > > > you mean "every" in the sense of "all" of set theory?

>
> > > > The latter is wrong, the former is correct.
> > > > (Note also every potentially infinite sequence only consist of finite
> > > > initial segments.)

>
> > > Let L be the potentially infinite
> > > list of natural numbers

>
> > > 1
> > > 2
> > > 3
> > > ...

>
> > > Does L have the property that
> > > every (in the sense of "all from 1 to n")
> > > natural number is a line of L

>
> > Yes.
>
> Let a potentially infinite set X be
> p-unlistable if
>     L is a potentially infinite
>     list of x's implies that L
>     does not have the property
>     that every (in the sense of
>     "all from 1 to n") x is a row of L
>
> A potentially infinite set Y is
> p-listable if it is not p-unlistable.
>
> We can divide the collection of
> potentially infinite sets into three
>
>     a:  p-unlistable potentially infinite
>         sets
>
>     b:  p-listable potentially infinite
>         sets
>
>     c:  potentially infinite sets that cannot
>         be shown to be p-listable or p-unlistable
>
> An example of a: is the potentially infinite
> set of all 0/1 sequences.
> An example of b: is the potentially infinite
> set of natural numbers.
> We do not have an example of c:  (there may
> not be one).-


You aked:
> > > > > > Can a potentially infinite list, L,
> > > > > > of potentially infinite 0/1 sequences
> > > > > > have the property that every
> > > > > > potentially infinite 0/1 sequence
> > > > > > is a line of L?

I answered yes (above: The latter is wrong, the former is correct.)

Now you claim the set of infinite 0/1 sequences is unlistable?

Regards, WM