Natural numbers and vases

[William]

Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_p a set of "dark" elements. Start off with no elements in vase 1 and each element of |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1. There is a set n iff n is in N_p.. At every step n, vase 1 contains a finite set of elements and vase 2 contains an infinite subset of N_p and N_d. We do not need N_d to insure that vase 2 contains and infinite number of elements
Note we never touch N_d so each element of N_d remains in vase 2.

We now continue until each element of N_p is in vase 1, (This takes an infinite number of steps.) At this point vase 1 contains each element of N_p, thus as infinite number of elements and vase 2 contains the empty set (a subset of N_p) and N_d. Note that N_d just sits there. We do not need N_d for anything.

Take the axiom of infinity. The the natural numbers are a Peano set and as a Peano set cannot contain a "dark" element N_d must be empty. So |N=|N_p


William Hughes

[Ross A. Finlayson]

So, what's the "supertask" that "takes infinite time"?

It seems that "in time" the supertask either "takes less time for each step
and in the limit sums to a finite time" or "no, never completes", or that
"there is a supertask for constant infinitesimals that results infinitely many
in 1.0s".

I.e. you seem to apply at least a model of real numbers, to time, to finish
in time (any "count" of time).

[WM]

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_d a set of "dark" elements. Start off with no elements in vase 1 and each element of |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1. There is a set n iff n is in N_p.. At every step n, vase 1 contains a finite set of elements and vase 2 contains an infinite subset of N_p and N_d. We do not need N_d to insure that vase 2 contains and infinite number of elements

Vase 2 contains an infinite number of elements as long as you remove only definable elements n. You will never remove more:
∀n ∈ ℕ_P: |ℕ \ {1, 2, 3, ..., n}| = ℵo (*)
But if you remove all elements collectively (which you fail to remove individually) then vase 2 is empty:
|ℕ \ {1, 2, 3, ...}| = 0 .

> Note we never touch N_d so each element of N_d remains in vase 2.

They cannot be touched individually.

>
> We now continue until each element of N_p is in vase 1, (This takes an infinite number of steps.)

Not even that would be sufficient because by definition (*). To remove all individually would require a last individual, and a next to the last and so on.

> At this point vase 1 contains each element of N_p, thus as infinite number of elements and vase 2 contains the empty set (a subset of N_p) and N_d. Note that N_d just sits there. We do not need N_d for anything.

We do need it for extending N_p as far as we like without losing the infinite buffer.

>
> Take the axiom of infinity. The the natural numbers are a Peano set and as a Peano set cannot contain a "dark" element N_d must be empty. So |N=|N_p
>

Every element of N_p can be removed individually. But there is no last one. And there is the condition (*) valid for every element of N_p.

Regards, WM

[William]

... a last individual, and a next to the last and so on.

"Let's start at the very beginning. A very good place to start"
There us a subset of N_p with no "last individual" (Follows trivially from the fact that N_p is a Peano set)

--
William Hughes

[Ross A. Finlayson]

I try and keep it for the vase and balls, that,
"there is only picking out a random ball, or,
pouring out balls at random", that any "balls
put in the vase, with a label", don't necessarily
come out in the same order, out, the labels, in.

Then algorithms in random selection result
"to retrieve a ball from the vase, retrieve a
random ball until it is the given ball", not even
saying whether samples are _replaced_ in
terms of them being random samples, under access:
this is for example a usual data structure with "most
first" when it works out that membership test, is, for
"components: principal, major, minor, and the tail",
why random selection results as enforcing why algorithms
have their terms, the asymptotic, or exhaustive.

One might aver "the balls besides the vase,
are marbles in a bounds, that can be seen at once".

It's not necessary "real numbers" only limits or
exhaustion - what results why supertasks complete
as explaining "summed up the infinite parts, sum".

[Fritz Feldhase]

On Sunday, May 1, 2022 at 10:43:47 PM UTC+2, WM wrote:

> Vase 2 contains an infinite number of elements as long as you remove only [finitely many] elements[:]
>
> ∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo

Indeed!

Though the formalisation of your claim is not quite correct. You might adopt the following one:

∀n ∈ ℕ: |ℕ \ {k_1, k_2, k_3, ..., k_n}| = ℵo ,

where (k_i) is defined with k_i e IN for all i e IN.

Hint: infinite - finite = infinite .

> But if you remove all elements [...] then vase 2 is empty:
>
> |ℕ \ ℕ| = 0 .

Indeed!

Hint: For integers x: x - x = 0, for sets X: X \ X = {}.

Well done, Mückenheim!

[WM]

How can that be? What would it imply?

No last number exists if beyond every number, which hitherto appears to be the last one, we can create new numbers.

But 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. If this is infinite for every n, then we have an infinite gap between all n and omega. This gap cannot be bridged because every n has the same infinite distance.

On the other hand, this gap can be bridged. We know it from

ℕ \ {1, 2, 3, ...} = { } .

Regards, WM

[sergio]

Whoa! "hitherto appears to be the lastist one"


WM has a real fear of being the Last One in line.

Perhaps he cuts in line to feel safer, to the detriment of n+1,n+2,... , which he turns his back on...

[William]

On Monday, May 2, 2022 at 9:11:30 AM UTC-3, WM wrote:

> gap

Nope, There is no "gap". The fact that every element of N_p is followed by a subset of N_p with cardinality aleph_0 does not mean there is a "gap" between N_p and omega. Cardinality is not a metric.

--
William Hughes

[WM]

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.

What do you not understand?

If this is infinite for every n, then we have an infinite gap between all n and omega.
This gap cannot be bridged because every n has the same infinite distance.

> The fact that every element of N_p is followed by a subset of N_p with cardinality aleph_0 does not mean there is a "gap" between N_p and omega.

No, but the fact that omega - n infinite proves that there is a gap.

> Cardinality is not a metric.

since cardinality is nonsense. But if omega exists and all n satisfy omega - n > 100, then there is a gap before omega.

Regards, WM

[sergio]

On 5/2/2022 3:14 PM, 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.

yes, like this omega - n = omega

>
> What do you not understand?

there is no gap

>
> If this is infinite for every n, then we have an infinite gap between all n and omega.

no, there is no gap. That is called an Endsegment

> This gap cannot be bridged because every n has the same infinite distance.

wrong. there is no gap, it is filled with natural numbers.

>
>> The fact that every element of N_p is followed by a subset of N_p with cardinality aleph_0 does not mean there is a "gap" between N_p and omega.
>
> No, but the fact that omega - n infinite proves that there is a gap.

Wrong, you misunderstand infinity.

>
>> Cardinality is not a metric.
>
> since cardinality is nonsense. But if omega exists and all n satisfy omega - n > 100, then there is a gap before omega.

still wrong, a lost little sheep that wants to be counted

>
> Regards, WM

[William]

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).

--
William Hughes

[Chris M. Thomasson]

On 5/1/2022 8:43 AM, William wrote:
>
> Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_p a set of "dark" elements. Start off with no elements in vase 1 and each element of |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1. There is a set n iff n is in N_p.. At every step n, vase 1 contains a finite set of elements and vase 2 contains an infinite subset of N_p and N_d. We do not need N_d to insure that vase 2 contains and infinite number of elements
> Note we never touch N_d so each element of N_d remains in vase 2.
>
> We now continue until each element of N_p is in vase 1, (This takes an infinite number of steps.)


> At this point

Humm... If it takes an infinite number of steps to fill vase 1, then at
what point can you say its full?

[Fritz Feldhase]

On Tuesday, May 3, 2022 at 12:14:56 AM UTC+2, Chris M. Thomasson wrote:
> On 5/1/2022 8:43 AM, William wrote:
> >
> > Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_p a set of "dark" elements. Start off with no elements in vase 1 and each element of |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1. There is a set n iff n is in N_p.. At every step n, vase 1 contains a finite set of elements and vase 2 contains an infinite subset of N_p and N_d. We do not need N_d to insure that vase 2 contains and infinite number of elements
> > Note we never touch N_d so each element of N_d remains in vase 2.
> >
> > We now continue until each element of N_p is in vase 1, (This takes an infinite number of steps.)
> >
> > At this point
> Humm... If it takes an infinite number of steps to fill vase 1, then at
> what point can you say its full?

I guess, he's considering a "super task". We start at t = 0. At t = 1 - 1/2 the first number, 1, is moved to vase 1. At t = 1 - 1/3 the second number, 2, is moved, at t = 1 - 1/4 the third number, 3, is moved and so on. Actually, it's easy to show that (in this case) for each and every number n there's a certain time t < 1, namely 1 - 1/(n + 1) , where n is moved to vase 1. Hence at t = 1 each and every number (in N_p) has been moved to vase 1.

So I'd say at t = 1 it's reasonable to claim that /now/ vase 1 is filled with all elements in N_p.

Yes, the process of "moving the numbers" does not have a last step. On the other hand, which number has n o t been moved to vase 1 at t = 1? :-P

For each and every t < 1 there are still infinitely many numbers which have to be moved to vase 1, but at t = 1 the job has been done. Such is the power of an "unlimited" speed up of a process.

[William]

On Monday, May 2, 2022 at 7:14:56 PM UTC-3, Chris M. Thomasson wrote:
> On 5/1/2022 8:43 AM, William wrote:
> >
> > Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_p a set of "dark" elements. Start off with no elements in vase 1 and each element of |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1. There is a set n iff n is in N_p.. At every step n, vase 1 contains a finite set of elements and vase 2 contains an infinite subset of N_p and N_d. We do not need N_d to insure that vase 2 contains and infinite number of elements
> > Note we never touch N_d so each element of N_d remains in vase 2.
> >
> > We now continue until each element of N_p is in vase 1, (This takes an infinite number of steps.)
>
>
> > At this point
> Humm... If it takes an infinite number of steps to fill vase 1, then at
> what point can you say its full?

There is no step at which the task is done (there is no step omega). We do not need to define a state for the vases after all steps are done. However, whether or not we define such a state it is clear that N_d remains in vase 2 (i,e, we do not need N_d). It remains true that N_p cannot contain a "dark" element so we never move a "dark" element from vase 2 to vase 1.

--
William Hughes

[Chris M. Thomasson]

I see. It seems to be akin to me writing the word "pi". I mean pi in
_all_ of it's infinite glory; t = 1, 100% complete. Not, t = .001, or t
= .5... I mean pi as in t = 1. ;^)

pi means pi with infinite precision in a sense. Does that sound kosher
at all?

I suppose I can write something like pi(5) which means 5 digits of pi
where pi(5) = 3.1415, and p(1) = 3, ect.. ;^)

[FromTheRafters]

Chris M. Thomasson brought next idea :

> On 5/1/2022 8:43 AM, William wrote:
>>
>> Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_p a set of
>> "dark" elements. Start off with no elements in vase 1 and each element of
>> |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1.
>> There is a set n iff n is in N_p.. At every step n, vase 1 contains
>> a finite set of elements and vase 2 contains an infinite subset of N_p and
>> N_d. We do not need N_d to insure that vase 2 contains and infinite number
>> of elements
>> Note we never touch N_d so each element of N_d remains in vase 2.
>>
>> We now continue until each element of N_p is in vase 1, (This takes an
>> infinite number of steps.)
>
>
>> At this point
>
> Humm... If it takes an infinite number of steps to fill vase 1, then at what
> point can you say its full?

When it is *all* done. :)

[sergio]

On 5/2/2022 6:43 PM, Fritz Feldhase wrote:
> On Tuesday, May 3, 2022 at 12:14:56 AM UTC+2, Chris M. Thomasson wrote:
>> On 5/1/2022 8:43 AM, William wrote:
>>>
>>> Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_p a set of "dark" elements. Start off with no elements in vase 1 and each element of |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1. There is a set n iff n is in N_p.. At every step n, vase 1 contains a finite set of elements and vase 2 contains an infinite subset of N_p and N_d. We do not need N_d to insure that vase 2 contains and infinite number of elements
>>> Note we never touch N_d so each element of N_d remains in vase 2.
>>>
>>> We now continue until each element of N_p is in vase 1, (This takes an infinite number of steps.)
>>>
>>> At this point
>> Humm... If it takes an infinite number of steps to fill vase 1, then at
>> what point can you say its full?
>
> I guess, he's considering a "super task". We start at t = 0. At t = 1 - 1/2 the first number, 1, is moved to vase 1. At t = 1 - 1/3 the second number, 2, is moved, at t = 1 - 1/4 the third number, 3, is moved and so on. Actually, it's easy to show that (in this case) for each and every number n there's a certain time t < 1, namely 1 - 1/(n + 1) , where n is moved to vase 1. Hence at t = 1 each and every number (in N_p) has been moved to vase 1.
>
> So I'd say at t = 1 it's reasonable to claim that /now/ vase 1 is filled with all elements in N_p.
>
> Yes, the process of "moving the numbers" does not have a last step. On the other hand, which number has n o t been moved to vase 1 at t = 1? :-P

this points out some of the limitations of language (which is poor at describing science), there is a good chapter in "the nature of physical science"
on drawbacks of languages.

t<1 it's "moving"
t=>1 it's "moved"

if you stop at k, when t<1, like WM does, then it is "truncated" move.

[WM]

The difference either consists of natural numbers or it is a gap. Cantor: ω - n = ω.

Regards, WM

[WM]

William schrieb am Dienstag, 3. Mai 2022 um 03:04:55 UTC+2:
> On Monday, May 2, 2022 at 7:14:56 PM UTC-3, Chris M. Thomasson wrote:
> > On 5/1/2022 8:43 AM, William wrote:
> > >
> > > Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_p a set of "dark" elements. Start off with no elements in vase 1 and each element of |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1. There is a set n iff n is in N_p.. At every step n, vase 1 contains a finite set of elements and vase 2 contains an infinite subset of N_p and N_d. We do not need N_d to insure that vase 2 contains and infinite number of elements
> > > Note we never touch N_d so each element of N_d remains in vase 2.
> > >
> > > We now continue until each element of N_p is in vase 1, (This takes an infinite number of steps.)
> >
> >
> > > At this point
> > Humm... If it takes an infinite number of steps to fill vase 1, then at
> > what point can you say its full?
> There is no step at which the task is done (there is no step omega). We do not need to define a state for the vases after all steps are done. However, whether or not we define such a state it is clear that N_d remains in vase 2 (i,e, we do not need N_d).

The difference ω - n = ω for every definablePeano n. Therefore the dark numbers are required.

Regards, WM

[WM]

FromTheRafters schrieb am Dienstag, 3. Mai 2022 um 13:08:28 UTC+2:

> >> At this point
> >
> > Humm... If it takes an infinite number of steps to fill vase 1, then at what
> > point can you say its full?
> When it is *all* done. :)

Then there remain ω - n = ω dark numbers nevertheless.

Regards, WM

[sergio]

no, the difference is natural numbers, because you stopped at k.

and no, you cannot "calculate" the difference, it is ω

>
> Regards, WM

WM, The Diversionator

[sergio]

not at all, they are the numbers in vase 2, and they are not dark.

Just pour them into vase 1 and be done with it.

[sergio]

so where are your dark numbers ? now you state there are Peano numbers ?

[Mostowski Collapse]

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:

[sergio]

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

[William]

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

[Fritz Feldhase]

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.

[Blass Nakae]

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?

[Jim Burns]

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.

[WM]

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

Either or! There is a dichotomy.

Regrads, WM

[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

[William]

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

[Gus Gassmann]

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.

Although the good prefosser is far too dense to understand, it easy to define a distance d on N u {omega}:
d(n,m) = |1/n - 1/m| if n,m =/= omega; d(n,omega) = 1/n.

[Fritz Feldhase]

> d(n,m) = |1/n - 1/m| if n,m =/= omega; d(n,omega) = [ d(omega,n) = ] 1/n [if n =/= omega]

and d(omega,omega) = 0, I guess. :-P

Nice.

[Jim Burns]

On 5/3/2022 4:45 PM, WM wrote:
> 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.

We know that,
if
each non-empty sub-collection of things =< β

contains first and last things,

then
each non-empty sub-collection of things =< β+1

contains first and last things...

...because we know that,
for each sub-collection C of things =< β+1,
C\{β+1} is a sub-collection of things =< β,

and,
if C\{β+1} has first and last things,
then C has first and last things.


We know that,
for each β, β < ω, for which

each non-empty sub-collection of things =< β

contains first and last things,

β+1 exists, β < β+1 < ω, for which

each non-empty sub-collection of things =< β+1

contains first and last things,

>> 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.

If there was some β for which

each non-empty sub-collection of things =< β

contains first and last things,

then that observation would lead to contradictions.

However, there is no such β

Assuming there is such β leads to contradictions.

[sergio]

On 5/3/2022 3:45 PM, WM wrote:
> 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.

there is no "touching", no "next to", no "gap".

Deal with it.

>>
>> 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.

that is your own shuffling switcharoo matrix of confusion you are stuck in.

>
> Regards, WM

[Gus Gassmann]

I take no credit for it. It's a pretty standard example.

[Ross A. Finlayson]

What _conditions_, remove, _contradictions_.

I.e., "relating the numbers otherwise to this-and-such
these others, how are both a sym-metric view between
zero and infinity, and, the unending view of zero to infinity,
both together according to the condition and guarantee -
these are simply exhaustion results either way".

[sergio]

Agree, Never shuffle in the Matrix.

[zelos...@gmail.com]

There is no such implication because YOU CANNOT EVEN FUCKING DEFINE WHAT A DARK NUMBER IS!

[WM]

There is a distance, if omega and omega + 1 and 2*omega etc. are existing.

> There is no distance. There is a "difference" but it is a set difference not a distance.

A difference in arithmetic is called a distance in geometry. There is an ordinal number sequece and thereferefor an ordinlal numbers axis. Before omega there is either nothing, i.e., a gap, or there is something. But this cannot be defined.

Regards, WM

[WM]

The observation that the X canot cover all the matrix is
1) simply a geometrical one and
2) provable in case of Cantors enumeration.

It is in contradiction with Cantors claim that the X could cover all the matrix.

Regards, WM

[WM]

zelos...@gmail.com schrieb am Mittwoch, 4. Mai 2022 um 06:54:32 UTC+2:
> tisdag 3 maj 2022 kl. 15:50:05 UTC+2 skrev WM:

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

> There is no such implication because YOU CANNOT EVEN FUCKING DEFINE WHAT A DARK NUMBER IS!

Either there is nothing immediately before omega, or it is dark.

Regards, WM

[Gus Gassmann]

On Wednesday, 4 May 2022 at 10:53:25 UTC-3, WM wrote:
[...]

> A difference in arithmetic is called a distance in geometry. There is an ordinal number sequece and thereferefor an ordinlal numbers axis. Before omega there is either nothing, i.e., a gap, or there is something. But this cannot be defined.

Waffling bullshit. If that is all you can contribute, get lost.

[Gus Gassmann]

As he said, "YOU CANNOT EVEN FUCKING DEFINE WHAT A DARK NUMBER IS!" All you have is "either there is something, or there is nothing". Well, whoop-de-doodoo!

[Fritz Feldhase]

On Wednesday, May 4, 2022 at 3:59:46 PM UTC+2, WM wrote:

> Either there is nothing immediately before omega, or it is dark.

There is nothing "immediately before" omega.

Hint: On the real number line there is nothing "immediately before" 1, still there is no "gap" between the real numbers < 1 and 1. In the same way there is no "gap" between the ordinal numbers < omega and omega.

Actually,

{x e IR : x < 1} u {1} = {x e IR : x <= 1}

and

{x e ORD : x < omega} u {omega} = {x e ORD : x <= omega}.

[sergio]

False Dichotomy

https://en.wikipedia.org/wiki/False_dilemma

[sergio]

Of course it is in contradiction with Cantors enumeration, because you are wrong.

>
> Regards, WM
>
>
>

[sergio]

On 5/4/2022 8:59 AM, WM wrote:

use your *cursor* function, it "defines" all numbers as it goes along the real line, no dark numbers ever!

[William]

On Wednesday, May 4, 2022 at 10:53:25 AM UTC-3, WM wrote:


> Before omega there is either nothing, i.e., a gap,

Nope, the word "gap" is only meaningful if there is a distance. The difference between omega and any element of N_p is a set difference, not a distance. The fact that this set difference has cardinality aleph_0, does not mean there is a distance which is infinite. There is no distance. The statement "omega -n = infinity" means "the cardinality of the set (omega\{1,2,3...,n}) is aleph_0". There is no largest ordinal that is smaller than omega. There is no "last" ordinal before omega'

==
William Hughes

[WM]

Fritz Feldhase schrieb am Mittwoch, 4. Mai 2022 um 16:52:13 UTC+2:
> On Wednesday, May 4, 2022 at 3:59:46 PM UTC+2, WM wrote:
>
> > Either there is nothing immediately before omega, or it is dark.
> There is nothing "immediately before" omega.
>
> Hint: On the real number line there is nothing "immediately before" 1, still there is no "gap" between the real numbers < 1 and 1.

This is not so easy to understand for you. Therefore I have used the ordinals. There is either something immediately before omega or nothing is immediately befor omega. The detrimental influence of matheology forbids even this simple logic.

Of course the same prevails before any defined real number, but it is not so obvious, because you can get as close as you like. At omega it is clear that omega - n = omega.

Regards, WM

[WM]

William schrieb am Mittwoch, 4. Mai 2022 um 17:45:14 UTC+2:
> On Wednesday, May 4, 2022 at 10:53:25 AM UTC-3, WM wrote:
>
>
> > Before omega there is either nothing, i.e., a gap,
> Nope,

Yes. Here we see the detrimental influence of matheology: denial of simplest logic.

If omega is a number larger than all natnumbers, then there is a difference which can be visualized as a distance.
This is the simplest obvious picture: 1, 2, 3, ..., ω, ω+1, ..., ω+k, 2ω, ...
It has a geometrical analogon.

> the word "gap" is only meaningful if there is a distance. The difference between omega and any element of N_p is a set difference, not a distance.

Wrong. "The system Omega in its natural order constitutes a 'sequence'." [Cantor, p. 445]

Regards, WM

[sergio]

On 5/4/2022 8:53 AM, WM wrote:
> William schrieb am Dienstag, 3. Mai 2022 um 23:01:47 UTC+2:
>> 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 a distance, if omega and omega + 1 and 2*omega etc. are existing.

A distance is a measure of length between two points.

what is your second point, and how are you measuring distance ?

>
>> There is no distance. There is a "difference" but it is a set difference not a distance.
>
> A difference in arithmetic is called a distance in geometry.

No. One is apples, the other is oranges. Google Vector Fields

> There is an ordinal number sequece

where ? which one ? or do you mean any sequence, {i,1,-0.3,17,n^n,} ?

> and thereferefor an ordinlal numbers axis.

Your words fail you.

> Before omega there is either nothing, i.e., a gap, or there is something. But this cannot be defined.

False Dichotomy

https://en.wikipedia.org/wiki/False_dilemma


>
> Regards, WM

keep looking in your gap, I am sure the dark ants will come out of it.

[WM]

William schrieb am Mittwoch, 4. Mai 2022 um 17:45:14 UTC+2:

> On Wednesday, May 4, 2022 at 10:53:25 AM UTC-3, WM wrote:
>
>
> > Before omega there is either nothing, i.e., a gap,
> Nope, the word "gap" is only meaningful if there is a distance. The difference between omega and any element of N_p is a set difference, not a distance.

That would be the case if only the statement "omega is different from every natnumber" were valid. But there is another statement: omega is larger than every natnumber. That is expressed in geometry by the ordinal line.

> There is no largest ordinal that is smaller than omega.

The definable ordinals are potentially infinite. There is no largest.

> There is no "last" ordinal before omega'

The dark ordinals are not fixable.

Regards, WM

[William]

Indeed. So what? The difference between two elements of a sequence may or may not be a distance. The difference between omega and any element of N_p is a set difference not a distance.

--
William Hughes

[William]

On Wednesday, May 4, 2022 at 1:11:40 PM UTC-3, WM wrote:

> potentially infinite.

Piffle.

--
William Hughes

[Jim Burns]

On 5/4/2022 9:58 AM, WM wrote:
> Jim Burns schrieb
> am Mittwoch, 4. Mai 2022 um 00:01:41 UTC+2:
>> On 5/3/2022 4:45 PM, WM wrote:

>>> Then explain please how the X's in
>>>
>>> XOOO...
>>> XOOO...
>>> XOOO...
>>> XOOO...
>>> ...
>>>
>>> can be shuffled to cover all positions of the matrix.
>>
>> If there was some β for which
>> each non-empty sub-collection of things =< β
>> contains first and last things,
>>
>> then that observation would lead to contradictions.
>
> The observation that the X canot cover all the matrix is

... incorrect.

Consider only
first-column X's at j/1 such that
each non-empty sub-collection of things =< j

contains first and last things,

and only
whole-matrix X's and O's at p/q such that
each non-empty sub-collection of things =< p and
each non-empty sub-collection of things =< q
contains first and last things.

We know that,
for each j,
there is one and only one pair p,q such that
j = (p+q-1)*(p+q-2)/2 + p

It is the pair p,q such that
p+q = ceiling((sqrt(8*j+1)+1)/2)
p = j - (p+q-1)*(p+q-2)/2
q = (p+q) - p

We know that,
for each pair p,q,
there is one and only one j such that
j = (p+q-1)*(p+q-2)/2 + p

We've gone over how we know this, and
you've accepted that much.
You simply reject that
the X's covering all the matrix
counts as
the X's covering all the matrix.

> The observation that the X canot cover all the matrix is
> 1) simply a geometrical one and

You do not type all the X's.
Less than all the X's do not cover all the matrix.
No one has said otherwise.

All the X's cover all the matrix.

> 2) provable in case of Cantors enumeration.
>
> It is in contradiction with Cantors claim that
> the X could cover all the matrix.

j = (p+q-1)*(p+q-2)/2 + p

[Jim Burns]

There is nothing which is immediately before omega.

For each j < omega,
j+1 exists, j < j+1 < omega.

For each j < omega,
j is not immediately before omega.

[WM]

It is an infinite number, in geometry, an infinite distance.

Regards, WM

[Jim Burns]

On 5/4/2022 9:53 AM, WM wrote:

> A difference in arithmetic is called a distance
> in geometry.

In geometry of the usual (Archimedean) variety,
for any two positive distances x and y,
there is a _finite ordinal_ n such that n*x > y.

For any _finite ordinal_ n,
each non-empty sub-collection of things =< n

contains first and last things.

> There is an ordinal number sequece and thereferefor
> an ordinlal numbers axis.

There is room in a geometric line for any finite
number of finite-length extensions.

There _isn't_ room in a geometric line for omega-many
(meaning aleph_0) finite-length extensions.

> Before omega there is either nothing, i.e., a gap,
> or there is something. But this cannot be defined.

For each thing before omega,
there is room for at least one more before omega.
Therefore, nothing before omega is
immediately before omega.

omega is not a reallyreallyreallyreallyreallyreally
large number. omega is a different kind of thing.

[sergio]

On 5/4/2022 10:48 AM, WM wrote:
> Fritz Feldhase schrieb am Mittwoch, 4. Mai 2022 um 16:52:13 UTC+2:
>> On Wednesday, May 4, 2022 at 3:59:46 PM UTC+2, WM wrote:
>>
>>> Either there is nothing immediately before omega, or it is dark.
>> There is nothing "immediately before" omega.
>>
>> Hint: On the real number line there is nothing "immediately before" 1, still there is no "gap" between the real numbers < 1 and 1.
>
> This is not so easy to understand for you. Therefore I have used the ordinals. There is either something immediately before omega or nothing is immediately befor omega. The detrimental influence of matheology forbids even this simple logic.

red herring. Your presentation assumes you have students that know nothing about math, and you then intentionally mislead them.

>
> Of course the same prevails before any defined real number, but it is not so obvious, because you can get as close as you like. At omega it is clear that omega - n = omega.

so what about omega + n ?



>
> Regards, WM
>
>

[sergio]

a simple proof the "professor" will not follow, as it proves there is no dark numbers.

[William]

Nope. Your calling it a distance does not make it so. You can call a set difference an "infinite distance" but that does not make it a distance.
>
> Regards, WM

[WM]

William schrieb am Mittwoch, 4. Mai 2022 um 19:55:02 UTC+2:
> On Wednesday, May 4, 2022 at 2:08:16 PM UTC-3, WM wrote:
> > William schrieb am Mittwoch, 4. Mai 2022 um 18:38:36 UTC+2:
> > > On Wednesday, May 4, 2022 at 1:00:22 PM UTC-3, WM wrote:
> > > > William schrieb am Mittwoch, 4. Mai 2022 um 17:45:14 UTC+2:
> > >
> > > > > the word "gap" is only meaningful if there is a distance. The difference between omega and any element of N_p is a set difference, not a distance.
> > > > Wrong. "The system Omega in its natural order constitutes a 'sequence'." [Cantor, p. 445]
> > > Indeed. So what? The difference between two elements of a sequence may or may not be a distance.

The difference between two ordinal numbers is a distance.
omega lies on the ordinal line as zero does.

> The difference between omega and any element of N_p is a set difference not a distance.

That is your personal and inconsistent definition.

Es fragt sich, in welchem Abstande von gamma dieser Gigant delta liegt
(Abstand = Distance)
1886, 11. Oct. Cantor, letter to Goldscheider

> > It is an infinite number, in geometry, an infinite distance.
> Nope. Your calling it a distance does not make it so.

An idea has no distance from a melody, but |a - b| is a distance for all ordinals a and b.

> You can call a set difference an "infinite distance" but that does not make it a distance.

That is your personal and inconsistent definition, by the way it is contradicting Cantor:

Es fragt sich, in welchem Abstande von gamma dieser Gigant delta liegt
(Abstand = Distance)
11. Oct 1886, Cantor, letter to Goldscheider

Regards, WM

[Gus Gassmann]

On Wednesday, 4 May 2022 at 14:55:02 UTC-3, William wrote:
> On Wednesday, May 4, 2022 at 2:08:16 PM UTC-3, WM wrote:

[...]

> > It is an infinite number, in geometry, an infinite distance.
> Nope. Your calling it a distance does not make it so. You can call a set difference an "infinite distance" but that does not make it a distance.

Much as I hate to give it to him, but the function f:N u {omega} -> N u {omega} defined by

f(n,m) = |n-m|, for all n,m in N
f(n,omega) = f(omega,n) = oo, for all n in N
f(omega,omega) = 0

is a metric:

f(x,x) = 0 for all x \in N u {omega}
f(x,y) = f(y,x) for all x,y \in N u {omega}
f(x,z) <= f(x,y) + f(y,z) for all x,y,z \in N u {omega}

Of course it doesn't prove anything regarding dark matter, but that's another story.

[WM]

Jim Burns schrieb am Mittwoch, 4. Mai 2022 um 19:39:10 UTC+2:
> On 5/4/2022 9:53 AM, WM wrote:
>
> > A difference in arithmetic is called a distance
> > in geometry.
> In geometry of the usual (Archimedean) variety,
> for any two positive distances x and y,
> there is a _finite ordinal_ n such that n*x > y.

The usual is not Cantor's extension.

> There _isn't_ room in a geometric line for omega-many
> (meaning aleph_0) finite-length extensions.

There is even much more room.

Let the solutions of ω^x = x be the giants of the second number class
Es fragt sich, in welchem Abstande von γ dieser Gigant δ liegt.

(Abstand = Distance)
1886, 11. Oct. Cantor, letter to Goldscheider

> > Before omega there is either nothing, i.e., a gap,
> > or there is something. But this cannot be defined.
> For each thing before omega,
> there is room for at least one more before omega.
> Therefore, nothing before omega is
> immediately before omega.

Then nothing is immediately before omega. That is called a gap.

>
> omega is not a reallyreallyreallyreallyreallyreally
> large number.

omega is not very large. It is the smallest transfinite ordinal number. But it is really larger than every natnumber.

> omega is a different kind of thing.

No, it is an ordinal number, tiny compared to a giant.

1, 2, 3, ... < omega < the smallest giant.

Regards, WM

[WM]

Either there is something immediately before omega or there is nothing. These are the two alternatives.

Regards, WM

[Fritz Feldhase]

On Wednesday, May 4, 2022 at 9:14:10 PM UTC+2, WM wrote:
>
> Then nothing is immediately before omega. That is called a gap.

No, this is called "a fact", idiot.

[William]

On Wednesday, May 4, 2022 at 4:00:01 PM UTC-3, WM wrote:

> The difference between two ordinal numbers is a distance.

Nope, false.

-- William Hughes

[FromTheRafters]

There are no transfinite ordinals before omega, that is why they call
it the *first* transfinite ordinal.

[Fritz Feldhase]

On Wednesday, May 4, 2022 at 9:16:32 PM UTC+2, WM wrote:

> Either there is something immediately before omega or there is nothing immediately before omega.

[Jim Burns]

On 5/4/2022 3:14 PM, WM wrote:
> Jim Burns schrieb
> am Mittwoch, 4. Mai 2022 um 19:39:10 UTC+2:
>> On 5/4/2022 9:53 AM, WM wrote:

>>> Before omega there is either nothing, i.e., a gap,
>>> or there is something. But this cannot be defined.
>>
>> For each thing before omega,
>> there is room for at least one more before omega.
>> Therefore, nothing before omega is
>> immediately before omega.
>
> Then nothing is immediately before omega.
> That is called a gap.

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

Nothing in discussion A is in that "gap".
No dark numbers.
A discussion with dark numbers
must be a different discussion.


For each j < omega,
each non-empty sub-collection of things =< j

contains first and last things.

We know that
no _last_ λ exists for which
each non-empty sub-collection of things =< λ

contains first and last things.

omega doesn't enter into it.

Appending λ+1 to the things =< λ leaves
each non-empty sub-collection with the
same pre-appendment first and last
or with λ+1.
So, λ can't be last.

So, nothing < omega is _next to_ omega.

[Gus Gassmann]

On Wednesday, 4 May 2022 at 16:16:32 UTC-3, WM wrote:
[...]

> Either there is something immediately before omega or there is nothing. These are the two alternatives.

You are going gangbusters today. Congratulations on this epiphany. The operative word, of course, is "immediately".

[William]

On Wednesday, May 4, 2022 at 4:16:32 PM UTC-3, WM wrote:
>
> Either there is something immediately before omega or there is nothing.

There is no position "immediately before omega" so it makes no sense to ask if there is something there.

--
William Hughes

[WM]

omega is a point. The next natnumber is in infinite distance. What is in between? If there is nothing in between, then the distance to every n cannot be infinite. It is deplorable how confusing matheology acts upon brains.

Regards, WM

[WM]

That is called a gap.

Regards, WM

[WM]

Jim Burns schrieb am Mittwoch, 4. Mai 2022 um 21:59:00 UTC+2:
> On 5/4/2022 3:14 PM, WM wrote:
> > Jim Burns schrieb
> > am Mittwoch, 4. Mai 2022 um 19:39:10 UTC+2:
> >> On 5/4/2022 9:53 AM, WM wrote:
>
> >>> Before omega there is either nothing, i.e., a gap,
> >>> or there is something. But this cannot be defined.
> >>
> >> For each thing before omega,
> >> there is room for at least one more before omega.
> >> Therefore, nothing before omega is
> >> immediately before omega.
> >
> > Then nothing is immediately before omega.
> > That is called a gap.
> In discussion A,
> each non-empty collection of things
> contains a first thing.

In discussion logic and mathematics, there is either something before the point omega or there is nothing. The latter is called a gap.

>
> Nothing in discussion A is in that "gap".

Then drop that kind of discussion.

Regards, WM

[WM]

Say in finite distance.

Regards, WM

[Jim Burns]

On 5/4/2022 4:16 PM, WM wrote:
> Jim Burns schrieb
> am Mittwoch, 4. Mai 2022 um 21:59:00 UTC+2:
>> On 5/4/2022 3:14 PM, WM wrote:
>>> Jim Burns schrieb
>>> am Mittwoch, 4. Mai 2022 um 19:39:10 UTC+2:
>>>> On 5/4/2022 9:53 AM, WM wrote:

>>>>> Before omega there is either nothing, i.e., a gap,
>>>>> or there is something. But this cannot be defined.
>>>>
>>>> For each thing before omega,
>>>> there is room for at least one more before omega.
>>>> Therefore, nothing before omega is
>>>> immediately before omega.
>>>
>>> Then nothing is immediately before omega.
>>> That is called a gap.
>>
>> In discussion A,
>> each non-empty collection of things
>> contains a first thing.
>
> In discussion logic and mathematics,
> there is either something before the point omega
> or there is nothing.

For each thing before omega,

something else is between it and omega.

There is nothing _immediately_ before omega.

> The latter is called a gap.

Whatever you call it,
there is nothing _immediately_ before omega.

"Nothing" includes dark numbers among
things which aren't immediately before omega.

>> Nothing in discussion A is in that "gap".
>
> Then drop that kind of discussion.

What does it say about your "argument"
that you (WM) need to _change what we say_
in order to "win"?

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

j is _definable_ in discussion A iff

each non-empty sub-collection of things =< j

contains first and last things,

In discussion B,
each thing is definable in discussion A.

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


( Also,
( each thing in discussion A is followed by
( another thing in discussion A.

[sergio]

On 5/4/2022 11:00 AM, WM wrote:
> William schrieb am Mittwoch, 4. Mai 2022 um 17:45:14 UTC+2:

>> On Wednesday, May 4, 2022 at 10:53:25 AM UTC-3, WM wrote:
>>
>>
>>> Before omega there is either nothing, i.e., a gap,

>> Nope,
>
> Yes. Here we see the detrimental influence of matheology: denial of simplest logic.
>
> If omega is a number larger than all natnumbers, then there is a difference which can be visualized as a distance.
> This is the simplest obvious picture: 1, 2, 3, ..., ω, ω+1, ..., ω+k, 2ω, ...

here you have assigned ω to be a natural number, just like n, fail.

>
>> the word "gap" is only meaningful if there is a distance. The difference between omega and any element of N_p is a set difference, not a distance.
>
> Wrong. "The system Omega in its natural order constitutes a 'sequence'." [Cantor, p. 445]

Wrong. red herring, That statement does not support your math at all.

>
> Regards, WM
>

you are a Deceiver.

[sergio]

On 5/4/2022 1:59 PM, WM wrote:
> William schrieb am Mittwoch, 4. Mai 2022 um 19:55:02 UTC+2:
>> On Wednesday, May 4, 2022 at 2:08:16 PM UTC-3, WM wrote:
>>> William schrieb am Mittwoch, 4. Mai 2022 um 18:38:36 UTC+2:
>>>> On Wednesday, May 4, 2022 at 1:00:22 PM UTC-3, WM wrote:
>>>>> William schrieb am Mittwoch, 4. Mai 2022 um 17:45:14 UTC+2:
>>>>
>>>>>> the word "gap" is only meaningful if there is a distance. The difference between omega and any element of N_p is a set difference, not a distance.
>>>>> Wrong. "The system Omega in its natural order constitutes a 'sequence'." [Cantor, p. 445]
>>>> Indeed. So what? The difference between two elements of a sequence may or may not be a distance.
>
> The difference between two ordinal numbers is a distance.
> omega lies on the ordinal line as zero does.
>
>> The difference between omega and any element of N_p is a set difference not a distance.
>
> That is your personal and inconsistent definition.
>
> Es fragt sich, in welchem Abstande von gamma dieser Gigant delta liegt
> (Abstand = Distance)
> 1886, 11. Oct. Cantor, letter to Goldscheider
>
>>> It is an infinite number, in geometry, an infinite distance.
>> Nope. Your calling it a distance does not make it so.
>
> An idea has no distance from a melody, but |a - b| is a distance for all ordinals a and b.

Phantasie

>
>> You can call a set difference an "infinite distance" but that does not make it a distance.
>
> That is your personal and inconsistent definition, by the way it is contradicting Cantor:
>
> Es fragt sich, in welchem Abstande von gamma dieser Gigant delta liegt
> (Abstand = Distance)
> 11. Oct 1886, Cantor, letter to Goldscheider

another quote from the Troll Book.


>
> Regards, WM

[sergio]

you are wrong on a number of items.

You have already been given a simple proof, MANY TIMES, that proves there is no number 'next to', or 'just before' omega.

But you rant on, so we must assume that

1. you do not understand math

and/or

2. you are intentional troll.



Which is it? 1, 2, 1 and 2 ?

[sergio]

"gap" is the wrong word usage, gap implies there are no numbers there, which is wrong.

It is full of numbers, an infinite amount of numbers.

[sergio]

WM => gap => means no numbers => must be dark things

that is obviously wrong, as the gap is full of numbers, an infinity of them.

[William]

On Wednesday, May 4, 2022 at 5:11:43 PM UTC-3, WM wrote:
> William schrieb am Mittwoch, 4. Mai 2022 um 22:00:54 UTC+2:
> > On Wednesday, May 4, 2022 at 4:16:32 PM UTC-3, WM wrote:
> > >
> > > Either there is something immediately before omega or there is nothing.
> > There is no position "immediately before omega" so it makes no sense to ask if there is something there.
> omega is a point. The next natnumber is in infinite distance.

Nope. There is no "distance".

--
William Hughes

[Chris M. Thomasson]

On 5/3/2022 9:54 PM, zelos...@gmail.com wrote:
> tisdag 3 maj 2022 kl. 15:50:05 UTC+2 skrev WM:

>> William schrieb am Dienstag, 3. Mai 2022 um 03:04:55 UTC+2:
>>> On Monday, May 2, 2022 at 7:14:56 PM UTC-3, Chris M. Thomasson wrote:
>>>> On 5/1/2022 8:43 AM, William wrote:
>>>>>
>>>>> Let natural numbers.|N be {N_p N_d} with N_p a Peano set and N_p a set of "dark" elements. Start off with no elements in vase 1 and each element of |N in vase 2. At step n, we mover element n, from N_p in vase 2 to vase 1. There is a set n iff n is in N_p.. At every step n, vase 1 contains a finite set of elements and vase 2 contains an infinite subset of N_p and N_d. We do not need N_d to insure that vase 2 contains and infinite number of elements
>>>>> Note we never touch N_d so each element of N_d remains in vase 2.
>>>>>
>>>>> We now continue until each element of N_p is in vase 1, (This takes an infinite number of steps.)
>>>>
>>>>
>>>>> At this point
>>>> Humm... If it takes an infinite number of steps to fill vase 1, then at
>>>> what point can you say its full?
>>> There is no step at which the task is done (there is no step omega). We do not need to define a state for the vases after all steps are done. However, whether or not we define such a state it is clear that N_d remains in vase 2 (i,e, we do not need N_d).

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

>> Regards, WM

> There is no such implication because YOU CANNOT EVEN FUCKING DEFINE WHAT A DARK NUMBER IS!

A dark number is one he has not thought of yet? Or perhaps a dark number
is one he new then forgot? Say, 42? Oh, well I guess 42 is no longer
dark because I just wrote it down. Actually, I wonder if he should think
about purchasing the following encyclopedia:

https://youtu.be/rVtHrgdcvZA

;^)

[Gus Gassmann]

That you don't have a clue about distances and metrics is blindingly obvious to anyone who can read. Your drivel is noted for its utter lack of meaning.

[sergio]

On 5/4/2022 11:57 AM, Jim Burns wrote:
> On 5/4/2022 9:58 AM, WM wrote:
>> Jim Burns schrieb
>> am Mittwoch, 4. Mai 2022 um 00:01:41 UTC+2:
>>> On 5/3/2022 4:45 PM, WM wrote:
>
>>>> Then explain please how the X's in
>>>>
>>>> XOOO...
>>>> XOOO...
>>>> XOOO...
>>>> XOOO...
>>>> ...
>>>>
>>>> can be shuffled to cover all positions of the matrix.
>>>
>>> If there was some β for which
>>> each non-empty sub-collection of things =< β

>>> contains first and last things,
>>>

>>> then that observation would lead to contradictions.
>>
>> The observation that the X canot cover all the matrix is
>
> ... incorrect.
>
> Consider only
> first-column X's at j/1 such that

> each non-empty sub-collection of things =< j
> contains first and last things,
>

> and only
> whole-matrix X's and O's at p/q such that
> each non-empty sub-collection of things =< p  and
> each non-empty sub-collection of things =< q

> contains first and last things.
>

> We know that,
> for each j,
> there is one and only one pair p,q such that
> j  =  (p+q-1)*(p+q-2)/2 + p
>
> It is the pair p,q such that
> p+q  =  ceiling((sqrt(8*j+1)+1)/2)
> p  =  j - (p+q-1)*(p+q-2)/2
> q  =  (p+q) - p
>
> We know that,
> for each pair p,q,
> there is one and only one j such that
> j  =  (p+q-1)*(p+q-2)/2 + p
>
> We've gone over how we know this, and
> you've accepted that much.
> You simply reject that
> the X's covering all the matrix
> counts as
> the X's covering all the matrix.
>
>> The observation that the X canot cover all the matrix is
>> 1) simply a geometrical one and
>
> You do not type all the X's.
> Less than all the X's do not cover all the matrix.
> No one has said otherwise.
>
> All the X's cover all the matrix.
>
>> 2) provable in case of Cantors enumeration.
>>
>> It is in contradiction with Cantors claim that
>> the X could cover all the matrix.
>
> j = (p+q-1)*(p+q-2)/2 + p
>


each and every position in the matrix has a p index and a q index at all times.

j is another index that is calculated by p and q, and applies at all times.

j = (p+q-1)*(p+q-2)/2 + p


So each and every position in the matrix, all of them is indexed by p,q, and j at all times.

there are no X nor Os required at all.

No swaparoo is needed, that is diversion.

[sergio]

That is a great set to have! I have referred to the book of Beeper Numbers many times.

The set of Binary Numbers is HUGE! no indexing there though, they are just listed in order.

The Book of Reverse Polish Numbers is very cool...



Unfortunately the URL is dark for the online "Dark Number Generator"

[zelos...@gmail.com]

onsdag 4 maj 2022 kl. 15:59:46 UTC+2 skrev WM:

> zelos...@gmail.com schrieb am Mittwoch, 4. Mai 2022 um 06:54:32 UTC+2:
> > tisdag 3 maj 2022 kl. 15:50:05 UTC+2 skrev WM:
>
> > > The difference ω - n = ω for every definablePeano n. Therefore the dark numbers are required.

> > There is no such implication because YOU CANNOT EVEN FUCKING DEFINE WHAT A DARK NUMBER IS!

> Either there is nothing immediately before omega, or it is dark.
>
> Regards, WM

Omega is an ordinal you fucking retard. THere is no "Before" it, Omega-1 is meaningless.

You sitll have failed to define "dark"

[WM]

Jim Burns schrieb am Mittwoch, 4. Mai 2022 um 22:59:41 UTC+2:
> On 5/4/2022 4:16 PM, WM wrote:

> > In discussion logic and mathematics,
> > there is either something before the point omega
> > or there is nothing.
>
> For each thing before omega,
> something else is between it and omega.

Right. Each definable thing has many successors because of potential infinity.
Each undefinable thing cannot be put in order.

But each definable thing has infinite distamce from omega. This distance must be realized somehow. It is existing as empty or as filled by dark numbers.

>
> There is nothing _immediately_ before omega.

How far from omega ist the first definable point before omega? According to Cantor it has distance omega. This distance would not exist, if there was nothing. Nothig cannot span a distance.

> > The latter is called a gap.
> Whatever you call it,
> there is nothing _immediately_ before omega.

How would you know?

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

omega is a point on the extended real axis. The points there have distances. "The question is which distance from gamma this giant delta has." [11. Oct 1886, Cantor, letter to Goldscheider] delta and gamma are very large transfinite ordinals. If you defend Cantor's theory, then accept his rules pleeze. And don't delete the evidence against your opinion.

Regards, WM

[WM]

omega is a point on the extended real axis. The points there have distances. In arithmetic we have "a is smaller than b, the difference is b - a", in geometry the same is expressed by "a lies left of b, the distance is b - a. There is no chance to keep the first and to reject the second.

"The question is which distance from gamma this giant delta has." [11. Oct 1886, Cantor, letter to Goldscheider] delta and gamma are very large transfinite ordinals. If you defend Cantor's theory, then accept his rules pleeze. And don't delete the clear evidence against your opinion.

Regards, WM

[WM]

Chris M. Thomasson schrieb am Donnerstag, 5. Mai 2022 um 01:09:47 UTC+2:

> A dark number is one he has not thought of yet?

Many dark numbers can be made definable as individuals. But most dark numbers cannot be thought of individually.

Regards, WM

[WM]

horand....@gmail.com schrieb am Donnerstag, 5. Mai 2022 um 01:59:41 UTC+2:
> On Wednesday, 4 May 2022 at 17:17:59 UTC-3, WM wrote:
> > horand....@gmail.com schrieb am Mittwoch, 4. Mai 2022 um 21:59:06 UTC+2:
> > > On Wednesday, 4 May 2022 at 16:16:32 UTC-3, WM wrote:
> > > [...]
> > > > Either there is something immediately before omega or there is nothing. These are the two alternatives.
> > > You are going gangbusters today. Congratulations on this epiphany. The operative word, of course, is "immediately".
> > Say in finite distance.

> distances and metrics

omega is a point on the extended real axis. The points there have distances. In arithmetic we have "a is smaller than b, the difference is b - a", in geometry the same is expressed by "a lies left of b, the distance is b - a. There is no chance to keep the first and to reject the second.

"The question is which distance from gamma this giant delta has." [11. Oct 1886, Cantor, letter to Goldscheider] delta and gamma are very large transfinite ordinals. If you defend Cantor's theory, then accept his rules pleeze. At least try to understand it.

Regards, WM

[WM]

zelos...@gmail.com schrieb am Donnerstag, 5. Mai 2022 um 07:07:29 UTC+2:
> onsdag 4 maj 2022 kl. 15:59:46 UTC+2 skrev WM:
> > zelos...@gmail.com schrieb am Mittwoch, 4. Mai 2022 um 06:54:32 UTC+2:
> > > tisdag 3 maj 2022 kl. 15:50:05 UTC+2 skrev WM:
> >
> > > > The difference ω - n = ω for every definablePeano n. Therefore the dark numbers are required.
> > > There is no such implication because YOU CANNOT EVEN FUCKING DEFINE WHAT A DARK NUMBER IS!
> > Either there is nothing immediately before omega, or it is dark.
>

> Omega is an ordinal . THere is no "Before" it, Omega-1 is meaningless.

omega is a point on the extended real axis. The points there have distances. In arithmetic we have "a is smaller than b, the difference is b - a", in geometry the same is expressed by "a lies left of b, the distance is b - a. There is no chance to keep the first and to reject the second.

"The question is which distance from gamma this giant delta has." [11. Oct 1886, Cantor, letter to Goldscheider] delta and gamma are very large transfinite ordinals. If you defend Cantor's theory, then accept his rules pleeze. And don't delete the clear evidence against your opinion.

> You sitll have failed to define "dark"

Dark is all between omega and the definable natnumbers. That is an infinite amount of numbers since "omega - n = omega" [Cantor].

Regards, WM

[Gus Gassmann]

On Thursday, 5 May 2022 at 08:29:49 UTC-3, WM wrote:
[...]

> omega is a point on the extended real axis. The points there have distances. In arithmetic we have "a is smaller than b, the difference is b - a", in geometry the same is expressed by "a lies left of b, the distance is b - a. There is no chance to keep the first and to reject the second.

Not true in this generality. Look up "taxicab norm".

[Mostowski Collapse]

You should team up with Dan Christensen, his DC Proof
is perfect for dark elements. Maybe we will soon
see a new version of Gödels incompletness theorem.

[sergio]

On 5/5/2022 6:34 AM, WM wrote:
> horand....@gmail.com schrieb am Donnerstag, 5. Mai 2022 um 01:59:41 UTC+2:
>> On Wednesday, 4 May 2022 at 17:17:59 UTC-3, WM wrote:
>>> horand....@gmail.com schrieb am Mittwoch, 4. Mai 2022 um 21:59:06 UTC+2:
>>>> On Wednesday, 4 May 2022 at 16:16:32 UTC-3, WM wrote:
>>>> [...]
>>>>> Either there is something immediately before omega or there is nothing. These are the two alternatives.
>>>> You are going gangbusters today. Congratulations on this epiphany. The operative word, of course, is "immediately".
>>> Say in finite distance.
>> distances and metrics
>
> omega is a point on the extended real axis.

No. Not a linear scale. Why you keep telling whoppers ?


<snip crap>

[sergio]

On 5/5/2022 6:25 AM, WM wrote:
> Jim Burns schrieb am Mittwoch, 4. Mai 2022 um 22:59:41 UTC+2:
>> On 5/4/2022 4:16 PM, WM wrote:
>
>>> In discussion logic and mathematics,
>>> there is either something before the point omega
>>> or there is nothing.
>>
>> For each thing before omega,
>> something else is between it and omega.
>
> Right. Each definable thing has many successors because of potential infinity.

How would you know ? you have an error is every post!

>
> But each definable thing has infinite distamce from omega. This distance must be realized somehow. It is existing as empty or as filled by dark numbers.

your daffynition of "definable" is quackers, and for stupid people.

>>
>> There is nothing _immediately_ before omega.
>

>>> The latter is called a gap.
>> Whatever you call it,
>> there is nothing _immediately_ before omega.
>
> How would you know?
>
>> omega is defined to be
>> the first thing in discussion A which
>> is not in discussion B.
>

<snip non applicable quote>


>
> Regards, WM