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