Date: Jan 31, 2013 9:21 AM
Author: William Hughes
Subject: Re: Matheology § 203

On Jan 31, 2:54 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 31 Jan., 14:41, William Hughes <wpihug...@gmail.com> wrote:
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> > On Jan 31, 10:23 am, WM <mueck...@rz.fh-augsburg.de> wrote:
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> > > On 31 Jan., 10:04, William Hughes <wpihug...@gmail.com> wrote:
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> > > > On Jan 31, 9:48 am, WM <mueck...@rz.fh-augsburg.de> wrote:
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> > > > > On 30 Jan., 22:38, William Hughes <wpihug...@gmail.com> wrote:
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> > > > > > On Jan 30, 10:24 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
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> > > > > > > On 30 Jan., 22:14, William Hughes <wpihug...@gmail.com> wrote:
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> > > > > > > > On Jan 30, 6:06 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
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> > > > > > > > > On 30 Jan., 12:32, William Hughes <wpihug...@gmail.com> wrote:
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> > > > > > > > > > On Jan 30, 12:21 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
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> > > > > > > > > > > On 30 Jan., 12:02, William Hughes <wpihug...@gmail.com> wrote:
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> > > > > > > > > > > > Summary.  We have agreed that
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> > > > > > > > > > > > For a potentially infinite list L, the
> > > > > > > > > > > > antidiagonal of L is not a line of L.

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> > > > > > > > > > Do you agree with the statement
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> > > > > > > > > > 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.

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> > > > > > > > So now we have
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> > > > > > > > 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.

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> > > > > > This does not answer the question.  Please answer the question.-
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> > > > > 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?

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

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> > > > 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.

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> > We can divide the collection of
> > potentially infinite sets into three

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> >     a:  p-unlistable potentially infinite
> >         sets

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> >     b:  p-listable potentially infinite
> >         sets

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> >     c:  potentially infinite sets that cannot
> >         be shown to be p-listable or p-unlistable

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> > 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

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