Date: Jan 31, 2013 9:54 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 203

On 31 Jan., 15:21, William Hughes <wpihug...@gmail.com> wrote:
> On Jan 31, 2:54 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
>
>

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


Obviously not.
>
> A potentially infinite list, L,
> of potentially infinite 0/1 sequences
> can have the property that every
> potentially infinite 0/1 sequence
> is a line of L-


Would you say that a line that is not in the list is in the list?

Of course we can construct another line that is not in the list by
diagonalization. But in the same way we can construct a line number
that is not in the list by diagonalization:

...000000000x1
...00000000x02
...0000000x003
...

with xxx = 111..., say.
Since we have no actual infinity, the number ...111 has not a number
of places that is larger than all natural numbers. Hence, it is a
natural number, unknown though.

Regards, WM