Date: Jan 31, 2013 8:41 AM
Author: William Hughes
Subject: Re: Matheology § 203

On Jan 31, 10:23 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> 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.

>
> > > > > > > > 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

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> > > > > > > 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

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> > > > > > 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.-
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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?

>
> > > 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).