remember these ?

1950's seeing your toes wiggle in your shoes, while you got several life time doses of xrays ?

https://en.wikipedia.org/wiki/Shoe-fitting_fluoroscope

they are now using these to combat Foot Covid in India

On 5/3/2022 9:59 AM, Mostowski Collapse wrote:
> Do you mean an unordered pair {N_p N_d} or the
> union N_p u N_d. These are two different things.
> Otherwise I can recommend Dan Christensen,
>
> he is quite a prolific dark matter researcher with
> a fist full of logic, he first went after radonized water,
> then snake oil, he is now into Half-functions
>
> A Borjomi mineral water ad from 1929, advertising the water as "radioactive"
> https://en.wikipedia.org/wiki/Radioactive_quackery
>
> Do these Half-functions of Dan Christensen have
> some Half-life time. Do they still exist after 10 years?
>
> William schrieb am Sonntag, 1. Mai 2022 um 17:44:05 UTC+2:
>> Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_p a set of "dark" elements.

On Tuesday, May 3, 2022 at 10:47:56 AM UTC-3, WM wrote:
> William schrieb am Montag, 2. Mai 2022 um 23:38:48 UTC+2:
> > On Monday, May 2, 2022 at 5:15:01 PM UTC-3, WM wrote:
> > > William schrieb am Montag, 2. Mai 2022 um 19:29:53 UTC+2:
> > > > On Monday, May 2, 2022 at 9:11:30 AM UTC-3, WM wrote:
> > > >
> > > > > gap
> > > >
> > > > Nope, There is no "gap".
> > > If we have a complete set of natnumbers n all of which are smaller than the limit omega, then we can calculate the difference omega - n.
> > >
> > Piffle. The "difference" is a set difference not a distance (metric).
> >
> The difference either consists of natural numbers or it is a gap.

Nope, The fact that there is no natural number between the *set* N_p and ω does not mean there is a "gap" between the set N_p and ω.

--
William Hughes

On Tuesday, May 3, 2022 at 7:16:34 PM UTC+2, William wrote:
> On Tuesday, May 3, 2022 at 10:47:56 AM UTC-3, WM wrote:
> >
> > The difference either consists of natural numbers or it is a gap.
> >
> Nope, The fact that there is no natural number between the *set* N_p and ω does not mean there is a "gap" between the set N_p and ω.

Actually, the fact that there is *no* _ordinal number_ between the set N_p (which consists of ordinal numbers) and ω which is another ordinal number "means" that there's no gap (concerning ordinal numbers) between the set N_p and ω.

WM is just a demented old crank full of shit.

Mostowski Collapse wrote:

> Do you mean an unordered pair {N_p N_d} or the union N_p u N_d. These
> are two different things. Otherwise I can recommend Dan Christensen,

you beautiful?

On 5/3/2022 9:49 AM, WM wrote:

> The difference ω - n = ω for every definable Peano n.
> Therefore the dark numbers are required.

No.

ω is defined to be the first thing in discussion A
which is not in discussion B.

In discussion A,
each non-empty collection of things
contains a first thing.

In discussion A,
each thing is followed by another thing.

A is the discussion of _ordinals_

ω is an ordinal, since it is in discussion A.

Some of the things in discussion A are _definable_

β is a _definable ordinal_ iff
each non-empty sub-collection of things =< β
contains a first thing and a last thing.

B is the discussion of the _definable ordinals_
also known as the natural numbers.

Either everything in discussion A is in discussion B,
or something in discussion A is not in discussion B.

If everything in discussion A is in discussion B,
then, since everything in B is definable,
there are no dark numbers in discussion A.

If something in discussion A is not in discussion B,
then the collection of things in A which are not-in B
is not empty.

Since this is discussion A, there is a first thing
in that non-empty collection.

That existing first thing is ω.

We know that,
if
each non-empty sub-collection of things =< β
contains a first thing and a last thing,
then
each non-empty sub-collection of things =< β+1
contains a first thing and a last thing.

| ...because we know that,
| if C is a sub-collection of things =< β+1
| then C\{β+1} is a sub-collection of things =< β
| and we know about sub-collections of things =< β
| | And then we work out the details.

Thus, we know that,
if
β is definable ( == in discussion B),
then
β+1 is definable (== in discussion B).

If β < ω
then
β is definable
β+1 is definable
β+1 < ω
no λ exists for which λ+1 = ω

> The difference ω - n = ω for every definable Peano n.
> Therefore the dark numbers are required.

No.

Either
everything in discussion A is in discussion B
and no dark numbers exist,
or
something in discussion A is not in discussion B
and no dark numbers exist.

In summary,
no dark numbers exist.

William schrieb am Dienstag, 3. Mai 2022 um 19:16:34 UTC+2:
> On Tuesday, May 3, 2022 at 10:47:56 AM UTC-3, WM wrote:
> > William schrieb am Montag, 2. Mai 2022 um 23:38:48 UTC+2:
> > > On Monday, May 2, 2022 at 5:15:01 PM UTC-3, WM wrote:
> > > > William schrieb am Montag, 2. Mai 2022 um 19:29:53 UTC+2:
> > > > > On Monday, May 2, 2022 at 9:11:30 AM UTC-3, WM wrote:
> > > > >
> > > > > > gap
> > > > >
> > > > > Nope, There is no "gap".
> > > > If we have a complete set of natnumbers n all of which are smaller than the limit omega, then we can calculate the difference omega - n.
> > > >
> > > Piffle. The "difference" is a set difference not a distance (metric).
> > >
> > The difference either consists of natural numbers or it is a gap.
> Nope, The fact that there is no natural number between the *set* N_p and ω does not mean there is a "gap" between the set N_p and ω.

Then there is no gap. Then there exists an element of N_p touching ω.

Either or! There is a dichotomy.

Regrads, WM

Jim Burns schrieb am Dienstag, 3. Mai 2022 um 21:18:17 UTC+2:
> On 5/3/2022 9:49 AM, WM wrote:
>
> > The difference ω - n = ω for every definable Peano n.
> > Therefore the dark numbers are required.
> No.

Either there is an n touching ω or there is a gap. No third alternative is possible.
>
> In summary,
> no dark numbers exist.

Then explain please how the X's in

XOOO...
XOOO...
XOOO...
XOOO...
....

can be shuffled to cover all positions of the matrix.

Regards, WM

On Tuesday, May 3, 2022 at 5:43:02 PM UTC-3, WM wrote:
> William schrieb am Dienstag, 3. Mai 2022 um 19:16:34 UTC+2:
> > On Tuesday, May 3, 2022 at 10:47:56 AM UTC-3, WM wrote:
> > > William schrieb am Montag, 2. Mai 2022 um 23:38:48 UTC+2:
> > > > On Monday, May 2, 2022 at 5:15:01 PM UTC-3, WM wrote:
> > > > > William schrieb am Montag, 2. Mai 2022 um 19:29:53 UTC+2:
> > > > > > On Monday, May 2, 2022 at 9:11:30 AM UTC-3, WM wrote:
> > > > > >
> > > > > > > gap
> > > > > >
> > > > > > Nope, There is no "gap".
> > > > > If we have a complete set of natnumbers n all of which are smaller than the limit omega, then we can calculate the difference omega - n.
> > > > >
> > > > Piffle. The "difference" is a set difference not a distance (metric).
> > > >
> > > The difference either consists of natural numbers or it is a gap.
> > Nope, The fact that there is no natural number between the *set* N_p and ω does not mean there is a "gap" between the set N_p and ω.
> Then there is no gap. Then there exists an element of N_p touching ω..

Piffle. You keep using words like "gap" and "touching" that only make sense if there is a distance. There is no distance. There is a "difference" but it is a set difference not a distance.

--
William Hughes

On Tuesday, 3 May 2022 at 18:01:47 UTC-3, William wrote:
[...]
> Piffle. You keep using words like "gap" and "touching" that only make sense if there is a distance. There is no distance. There is a "difference" but it is a set difference not a distance.