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.

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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

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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?

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> > > > > 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

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> > 1

> > 2

> > 3

> > ...

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> > 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).