lørdag 8. januar 2022 kl. 12:35:16 UTC+1 skrev WM:
> Fritz Feldhase schrieb am Freitag, 7. Januar 2022 um 13:39:42 UTC+1:
> > On Friday, January 7, 2022 at 12:08:45 PM UTC+1, WM wrote:
> >
> > > The dark natnumbers have no discernible order.
> > This proves that there are no dark numbers in IN, since IN is a totally orderd set [by the usual <=].
> What is the point on the real axis next to 0? There is something. There is no gap. What is it? It cannot be put in any order.
> >
> > You know, for EACH and EVERY element n in IN and EACH and EVERY element m in IN: Either n < m, m < n, or n = m.
> For all those elements subtraction leaves a big rest
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> But subtracting all elements
> |ℕ \ {1, 2, 3, ...}| = 0
> leaves nothing.
> Hence there is a difference between ℕ_def and ℕ.
> >
> > Hence there are no "dark numbers" in IN.
> There is a difference between ℕ_def and ℕ. This is dark.
>
> Regards, WM f |ℕ \ {1, 2, 3, ...}| = 0 leaves nothing, why don't |ℕ \ {1, 2, 3, ..., w}| = 0 leaves nothing?
KON

If f |ℕ \ {1, 2, 3, ...}| = 0 leaves nothing, why don't |ℕ \ {1, 2, 3, ..., w} leaves nothing?
w (omega) is a natural number.
Why is f|ℕ \ {1, 2, 3, ..., w }|= {w}
KON

William schrieb am Samstag, 8. Januar 2022 um 18:41:11 UTC+1:
> On Saturday, January 8, 2022 at 8:29:24 AM UTC-4, WM wrote:
> > William schrieb am Freitag, 7. Januar 2022 um 15:57:15 UTC+1:
> > > On Friday, January 7, 2022 at 7:08:45 AM UTC-4, WM wrote:
> >
> > > > > > > Each element that is in N_P is followed by an infinite number of elements that are in N_P.
> > > > > > If you take the union over [the set of] FISONs then ℵo elements remain outside.
> > > > >
> > > > > Nope. N_P is defined to be the union of the set of FISONS. There is no largest FISON. Thus there is no largest element of N_P
> > Fine. I call this ℕ_def.
> Nope. |N_def is the set of natural numbers that have the property they can be written down. |N_def is a proper subset of N_P .

What elements of N_P cannot be written down?

Regards, WM

It happens that WM formulated :
> William schrieb am Samstag, 8. Januar 2022 um 18:41:11 UTC+1:
>> On Saturday, January 8, 2022 at 8:29:24 AM UTC-4, WM wrote:
>>> William schrieb am Freitag, 7. Januar 2022 um 15:57:15 UTC+1:
>>>> On Friday, January 7, 2022 at 7:08:45 AM UTC-4, WM wrote:
>>>>>>>> Each element that is in N_P is followed by an infinite number of
>>>>>>>> elements that are in N_P.
>>>>>>> If you take the union over [the set of] FISONs then ℵo elements remain
>>>>>>> outside.
>>>>>>
>>>>>> Nope. N_P is defined to be the union of the set of FISONS. There is no
>>>>>> largest FISON. Thus there is no largest element of N_P
>>> Fine. I call this ℕ_def.
>> Nope. |N_def is the set of natural numbers that have the property they can
>> be written down. |N_def is a proper subset of N_P .
>
> What elements of N_P cannot be written down?

The complement set to the proper subset which only require finite
resources with which to do the writing.

On Sunday, January 9, 2022 at 8:58:37 AM UTC-4, WM wrote:
> William schrieb am Samstag, 8. Januar 2022 um 18:41:11 UTC+1:
> > On Saturday, January 8, 2022 at 8:29:24 AM UTC-4, WM wrote:
> > > William schrieb am Freitag, 7. Januar 2022 um 15:57:15 UTC+1:
> > > > On Friday, January 7, 2022 at 7:08:45 AM UTC-4, WM wrote:
> > >
> > > > > > > > Each element that is in N_P is followed by an infinite number of elements that are in N_P.
> > > > > > > If you take the union over [the set of] FISONs then ℵo elements remain outside.
> > > > > >
> > > > > > Nope. N_P is defined to be the union of the set of FISONS. There is no largest FISON. Thus there is no largest element of N_P
> > > Fine. I call this ℕ_def.
> > Nope. |N_def is the set of natural numbers that have the property they can be written down. |N_def is a proper subset of N_P .
>
> What elements of N_P cannot be written down?

Only a finite set of natural numbers can be written down. N_P, like any Peano set, is infinite.

--
William Hughes

>
> Regards, WM

lördag 8 januari 2022 kl. 12:35:16 UTC+1 skrev WM:
> Fritz Feldhase schrieb am Freitag, 7. Januar 2022 um 13:39:42 UTC+1:
> > On Friday, January 7, 2022 at 12:08:45 PM UTC+1, WM wrote:
> >
> > > The dark natnumbers have no discernible order.
> > This proves that there are no dark numbers in IN, since IN is a totally orderd set [by the usual <=].
> What is the point on the real axis next to 0? There is something. There is no gap. What is it? It cannot be put in any order.
> >
> > You know, for EACH and EVERY element n in IN and EACH and EVERY element m in IN: Either n < m, m < n, or n = m.
> For all those elements subtraction leaves a big rest
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> But subtracting all elements
> |ℕ \ {1, 2, 3, ...}| = 0
> leaves nothing.
> Hence there is a difference between ℕ_def and ℕ.
> >
> > Hence there are no "dark numbers" in IN.
> There is a difference between ℕ_def and ℕ. This is dark.
>
> Regards, WM

>What is the point on the real axis next to 0? There is something. There is no gap. What is it? It cannot be put in any order.

That is REAL numbers and just because there is no NEXT doesn't that mean there isn't an order you fucking moron.

>Hence there is a difference between ℕ_def and ℕ.

Non-sequitor.

your N_def=N

William schrieb am Sonntag, 9. Januar 2022 um 18:08:08 UTC+1:
> On Sunday, January 9, 2022 at 8:58:37 AM UTC-4, WM wrote:

> > What elements of N_P cannot be written down?
> Only a finite set of natural numbers can be written down.
Yes.
> N_P, like any Peano set, is infinite.
But all elements having distance ℵo from omega, form an not actually infinite set together.
∀n ∈ ℕ_P: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
Therefore ℕ_P is not actually infinite. What remains?

Regards, WM

On Tuesday, January 11, 2022 at 7:17:28 AM UTC+1, WM wrote:
> William schrieb am Sonntag, 9. Januar 2022 um 18:08:08 UTC+1:
> >
> > IN, like any Peano set, is infinite.
> >
> But all [natural numbers] having distance ℵo from omega, form an [...] infinite set.

Indeed!

card({n e IN : card(IN \ {1, 2, 3, ..., n}) = ℵo}) = ℵo..

> Therefore <bla>

Yeah, whatever.

On 1/11/2022 12:17 AM, WM wrote:
> William schrieb am Sonntag, 9. Januar 2022 um 18:08:08 UTC+1:
>> On Sunday, January 9, 2022 at 8:58:37 AM UTC-4, WM wrote:
>
>>> What elements of N_P cannot be written down?
>> Only a finite set of natural numbers can be written down.
> Yes.

what does "written down" mean mathematically ? Or is it simply a time delay to try to exclude infinite sets ?

>> N_P, like any Peano set, is infinite.
> But all elements having distance ℵo from omega,

Ok

>form an not actually infinite set together.

no.

On Tuesday, January 11, 2022 at 2:17:28 AM UTC-4, WM wrote:

> ∀n ∈ ℕ_P: |ℕ[_P] \ {1, 2, 3, ..., n}| = ℵo.

Each set of natural numbers {1,2,3,...,n} hast a last element.

> Therefore ℕ_P is not actually infinite.

Nope

>What remains?

A set of natural numbers with no last element.

--
William Hughes

On Tuesday, January 11, 2022 at 2:17:28 AM UTC-4, WM wrote:

> ∀n ∈ ℕ_P: |ℕ [_P]\ {1, 2, 3, ..., n}| = ℵo.

Note that each set {1, 2, 3, ..., n} is a proper subset of the natural numbers,

> Therefore ℕ_P is not actually infinite.

Nope

> What remains?

A set that is an improper subset of the natural numbers.

--
William Hughes

On Tuesday, 11 January 2022 at 13:19:13 UTC-4, William wrote:
> On Tuesday, January 11, 2022 at 2:17:28 AM UTC-4, WM wrote:
>
> > ∀n ∈ ℕ_P: |ℕ [_P]\ {1, 2, 3, ..., n}| = ℵo.
>
> Note that each set {1, 2, 3, ..., n} is a proper subset of the natural numbers,
> > Therefore ℕ_P is not actually infinite.
> Nope
>
> > What remains?
>
> A set that is an improper subset of the natural numbers.

I think you should explain what you mean by "proper" and "improper" subsets here. Near as an innocent bystander can decipher, "proper" means "finite", so "improper" is "not finite"? That's problematic, you know...

On Tuesday, January 11, 2022 at 3:59:25 PM UTC-4, horand....@gmail.com wrote:
> On Tuesday, 11 January 2022 at 13:19:13 UTC-4, William wrote:
> > On Tuesday, January 11, 2022 at 2:17:28 AM UTC-4, WM wrote:
> >
> > > ∀n ∈ ℕ_P: |ℕ [_P]\ {1, 2, 3, ..., n}| = ℵo.
> >
> > Note that each set {1, 2, 3, ..., n} is a proper subset of the natural numbers,
> > > Therefore ℕ_P is not actually infinite.
> > Nope
> >
> > > What remains?
> >
> > A set that is an improper subset of the natural numbers.
> I think you should explain what you mean by "proper" and "improper" subsets here. Near as an innocent bystander can decipher, "proper" means "finite", so "improper" is "not finite"? That's problematic, you know...

The terms "proper" and "improper" are standard terms. A subset R of T is a proper subset if it does not contain all the elements of T.
i.e. T\R is not the empty set. An improper subset contains all the elements of T. The only improper subset of T is T. A proper subset of N_P can be infinite, consider the even numbers. I use sets as "What remains" is not an element of S_P but a subset of S_P.

--
William Hughes

On Tuesday, 11 January 2022 at 19:16:39 UTC-4, William wrote:
> On Tuesday, January 11, 2022 at 3:59:25 PM UTC-4, horand....@gmail.com wrote:
> > On Tuesday, 11 January 2022 at 13:19:13 UTC-4, William wrote:
> > > On Tuesday, January 11, 2022 at 2:17:28 AM UTC-4, WM wrote:
> > >
> > > > ∀n ∈ ℕ_P: |ℕ [_P]\ {1, 2, 3, ..., n}| = ℵo.
> > >
> > > Note that each set {1, 2, 3, ..., n} is a proper subset of the natural numbers,
> > > > Therefore ℕ_P is not actually infinite.
> > > Nope
> > >
> > > > What remains?
> > >
> > > A set that is an improper subset of the natural numbers.
> > I think you should explain what you mean by "proper" and "improper" subsets here. Near as an innocent bystander can decipher, "proper" means "finite", so "improper" is "not finite"? That's problematic, you know...
> The terms "proper" and "improper" are standard terms. A subset R of T is a proper subset if it does not contain all the elements of T.
> i.e. T\R is not the empty set. An improper subset contains all the elements of T. The only improper subset of T is T. A proper subset of N_P can be infinite, consider the even numbers. I use sets as "What remains" is not an element of S_P but a subset of S_P.

Got it. I still think its a sneaky (and easily misunderstood) way to say that N_P = N

Gus Gassmann pretended :
> On Tuesday, 11 January 2022 at 19:16:39 UTC-4, William wrote:
>> On Tuesday, January 11, 2022 at 3:59:25 PM UTC-4, horand....@gmail.com
>> wrote:
>>> On Tuesday, 11 January 2022 at 13:19:13 UTC-4, William wrote:
>>>> On Tuesday, January 11, 2022 at 2:17:28 AM UTC-4, WM wrote:
>>>>
>>>>> ∀n ∈ ℕ_P: |ℕ [_P]\ {1, 2, 3, ..., n}| = ℵo.
>>>>
>>>> Note that each set {1, 2, 3, ..., n} is a proper subset of the natural
>>>> numbers,
>>>>> Therefore ℕ_P is not actually infinite.
>>>> Nope
>>>>
>>>>> What remains?
>>>>
>>>> A set that is an improper subset of the natural numbers.
>>> I think you should explain what you mean by "proper" and "improper" subsets
>>> here. Near as an innocent bystander can decipher, "proper" means "finite",
>>> so "improper" is "not finite"? That's problematic, you know...
>> The terms "proper" and "improper" are standard terms. A subset R of T is a
>> proper subset if it does not contain all the elements of T. i.e. T\R is not
>> the empty set. An improper subset contains all the elements of T. The only
>> improper subset of T is T. A proper subset of N_P can be infinite, consider
>> the even numbers. I use sets as "What remains" is not an element of S_P but
>> a subset of S_P.
>
> Got it. I still think its a sneaky (and easily misunderstood) way to say that
> N_P = N

Saying finite sets have a last element is another case like this where
the wording is such that it works for both finite and infinite sets.
Having or not having a last element is also a sneaky way to to say a
set is finite or infinite.

Even if 'improper' it were not easiliy misunderstood, WM would still
have trouble with it. He sees no difference between a set and its
elements.

FromTheRafters schrieb am Mittwoch, 12. Januar 2022 um 10:06:42 UTC+1:

> Even if 'improper' it were not easiliy misunderstood, WM would still
> have trouble with it. He sees no difference between a set and its
> elements.

There is a difference. The elements are many, the set is a single entity. But I dare to consider all elements without treating them as a set.
All elements which have ℵo elements between themselves and omega can be collected. By collecting them, this distance will not be extinguished. Matheologians claim that as soon as the elements are collected into a set, the distance has vanished. That is nonsense.

Regards, WM

William schrieb am Dienstag, 11. Januar 2022 um 18:19:13 UTC+1:
> On Tuesday, January 11, 2022 at 2:17:28 AM UTC-4, WM wrote:
>
> > ∀n ∈ ℕ_P: |ℕ [_P]\ {1, 2, 3, ..., n}| = ℵo.
>
> Note that each set {1, 2, 3, ..., n} is a proper subset of the natural numbers,
> > Therefore ℕ_P is not actually infinite.
> Nope
>
> > What remains?
>
> A set that is an improper subset of the natural numbers.

I dare to consider all its elements without treating them as a set. They have ℵo elements between themselves and omega. By collecting them, this distance will not be extinguished. Matheologians claim that as soon as the elements are collected into a set, the distance has vanished. That is nonsense.

Regards, WM

On 1/12/2022 2:57 PM, WM wrote:
> FromTheRafters schrieb am Mittwoch, 12. Januar 2022 um 10:06:42 UTC+1:
>
>> Even if 'improper' it were not easiliy misunderstood, WM would still
>> have trouble with it. He sees no difference between a set and its
>> elements.
>
> There is a difference. The elements are many, the set is a single entity. But I dare to consider all elements without treating them as a set.

....then the are not elements of a set.

> All elements which have ℵo elements between themselves and omega can be collected.

collected by the infinite vacuum cleaner, or an ENDSEGMENT, IF it is an ordered set of natural numbers at element k.

>By collecting them, this distance will not be extinguished.

what "distance" ? what "not extinguished" ?

> Matheologians claim that as soon as the elements are collected into a set, the distance has vanished. That is nonsense.

explain yourself...

>
> Regards, WM
>

On 1/12/2022 3:00 PM, WM wrote:
> William schrieb am Dienstag, 11. Januar 2022 um 18:19:13 UTC+1:
>> On Tuesday, January 11, 2022 at 2:17:28 AM UTC-4, WM wrote:
>>
>>> ∀n ∈ ℕ_P: |ℕ [_P]\ {1, 2, 3, ..., n}| = ℵo.
>>
>> Note that each set {1, 2, 3, ..., n} is a proper subset of the natural numbers,
>>> Therefore ℕ_P is not actually infinite.
>> Nope
>>
>>> What remains?
>>
>> A set that is an improper subset of the natural numbers.
>
> I dare to consider all its elements without treating them as a set.

then they are not elements of a set.

>They have ℵo elements between themselves and omega.

what set are you not talking about ?

> By collecting them,

by hand ?

? this distance will not be extinguished.

distances do not get extinguished.

> Matheologians claim that as soon as the elements are collected into a set, the distance has vanished. That is nonsense.

what you post is nonsence, you need context.

>
> Regards, WM
>

WM brought next idea :
> FromTheRafters schrieb am Mittwoch, 12. Januar 2022 um 10:06:42 UTC+1:
>
>> Even if 'improper' it were not easiliy misunderstood, WM would still
>> have trouble with it. He sees no difference between a set and its
>> elements.
>
> There is a difference. The elements are many, the set is a single entity. But
> I dare to consider all elements without treating them as a set.

So does everyone else, but if you want to have them in a set -- it is a
subset, you're not changing the superset. Even if you 'left elements
behind' in your consideration it is a subset, but then it could be
called a 'proper subset'. I'm glad you backed off from the idea that a
set is its elements and nothing more.

On Wednesday, January 12, 2022 at 5:00:44 PM UTC-4, WM wrote:

> I dare to consider all its elements without treating them as a set.

This does not matter. When you consider "all its elements" you do not consider an element that is larger than all the elements you are considering.

> They have ℵo elements between themselves and omega.

Correct, each of the elements you are considering has ℵo elements between itself and omega. Also correct is that if you consider an element of N_P, then this element is not larger than all the elements you are considering

> By collecting them, this distance will not be extinguished.

So you say. Since "distance" is your word I cannot dispute this. However, it remains true that if you consider an element of N_P you are considering something that is not larger than all the elements that you are considering.

--
William Hughes

On Tuesday, January 11, 2022 at 3:16:39 PM UTC-8, William wrote:
> On Tuesday, January 11, 2022 at 3:59:25 PM UTC-4, horand....@gmail.com wrote:
> > On Tuesday, 11 January 2022 at 13:19:13 UTC-4, William wrote:
> > > On Tuesday, January 11, 2022 at 2:17:28 AM UTC-4, WM wrote:
> > >
> > > > ∀n ∈ ℕ_P: |ℕ [_P]\ {1, 2, 3, ..., n}| = ℵo.
> > >
> > > Note that each set {1, 2, 3, ..., n} is a proper subset of the natural numbers,
> > > > Therefore ℕ_P is not actually infinite.
> > > Nope
> > >
> > > > What remains?
> > >
> > > A set that is an improper subset of the natural numbers.
> > I think you should explain what you mean by "proper" and "improper" subsets here. Near as an innocent bystander can decipher, "proper" means "finite", so "improper" is "not finite"? That's problematic, you know...
> The terms "proper" and "improper" are standard terms. A subset R of T is a proper subset if it does not contain all the elements of T.
> i.e. T\R is not the empty set. An improper subset contains all the elements of T. The only improper subset of T is T. A proper subset of N_P can be infinite, consider the even numbers. I use sets as "What remains" is not an element of S_P but a subset of S_P.
>
> --
> William Hughes

Eh, hrm, ....

I think people know that "the usual product of infinitely
many copies of N is _defined_ as the empty set, another
_defined_ as the set itself".

This is after matters of direct product , the direct products
and sums of sets like N and their number theories' notions
of compactness.

I.e. "N models a set of standard integers" while "number theory's
elements are natural".

Here that's where "there isn't a non-standard/extra-ordinary set
in regular set theory called N".

(For naturals.)

I think most people accept that the direct product is empty,
but it's as arrived at the other way, that most people who know it's
defined either do or don't know there's the corresponding
cases that result from it infinite both ways besides finite both ways.

So, defining the arithmetic of the finite and infinite in terms of each other,
is that the numbers are number theories', and for arithmetic.

It's not so much more involved why 0^0 is 0 or 1.
I.e. people who read it x^ 0+ read it 1, while , those
who read it 0^ x read it 0. It can only be one or the other -
the discontinuity removable in the either the graph of zero or
graph of x^0, it's one or the other and still a discontinuity,
with respect to the other.

The "N + N + N + ... = 0" or "N + N + N + ... = N", it's
one or the other or "an argument in the counterexample of topology".

Or number theory, ....

"That there is a point at infinity", is probably among number theory's
most "conjectured" or "defined", terms to establish linear regularity
for all the rest of them, the numbers. (Or as what does.)

After laws of arithmetic there are more of them,
here laws of small and large numbers in arithmetic.

FromTheRafters schrieb am Mittwoch, 12. Januar 2022 um 22:38:19 UTC+1:
> WM brought next idea :
> > FromTheRafters schrieb am Mittwoch, 12. Januar 2022 um 10:06:42 UTC+1:
> >
> >> Even if 'improper' it were not easiliy misunderstood, WM would still
> >> have trouble with it. He sees no difference between a set and its
> >> elements.
> >
> > There is a difference. The elements are many, the set is a single entity. But
> > I dare to consider all elements without treating them as a set.
> So does everyone else, but if you want to have them in a set -- it is a
> subset, you're not changing the superset.

It is the subset ℕ_def.
In particular I am not changing the remaining ℵo elements of ℕ \ ℕ_def. They are dark.

Regards, WM

William schrieb am Mittwoch, 12. Januar 2022 um 23:43:31 UTC+1:
> On Wednesday, January 12, 2022 at 5:00:44 PM UTC-4, WM wrote:
>
> > I dare to consider all its elements without treating them as a set.
> This does not matter.

It does. When all its elements have distance ℵo from omega, then the set has distance ℵo from omega.

> > By collecting them, this distance will not be extinguished.
> So you say. Since "distance" is your word I cannot dispute this.

That word means that between every element of the subset and omega, there are ℵo elements, ℵo of which cannot be defined. They do not belong to the subset N_P.

> However, it remains true that if you consider an element of N_P you are considering something that is not larger than all the elements that you are considering.

You can find a larger element in every case. But this larger element has same distance from omega.

Regards, WM

On Thursday, January 13, 2022 at 6:36:37 AM UTC+1, WM wrote:

> It is the subset ℕ_def.

Hint: IN_def = IN.

You know: An e IN: card({IN \ {1, 2, 3, ..., n}) = aleph_0.

Hence IN_def = {n e IN : card({IN \ {1, 2, 3, ..., n}) = aleph_0} = IN. qed

on 1/13/2022, WM supposed :
> FromTheRafters schrieb am Mittwoch, 12. Januar 2022 um 22:38:19 UTC+1:
>> WM brought next idea :
>>> FromTheRafters schrieb am Mittwoch, 12. Januar 2022 um 10:06:42 UTC+1:
>>>
>>>> Even if 'improper' it were not easiliy misunderstood, WM would still
>>>> have trouble with it. He sees no difference between a set and its
>>>> elements.
>>>
>>> There is a difference. The elements are many, the set is a single entity.
>>> But I dare to consider all elements without treating them as a set.
>> So does everyone else, but if you want to have them in a set -- it is a
>> subset, you're not changing the superset.
>
> It is the subset ℕ_def.
> In particular I am not changing the remaining ℵo elements of ℕ \ ℕ_def.

You are not changing *anything*, you are 'considering' the set
difference between two sets as a set. You are not subtracting from and
hence changing a superset.

*THEY DO NOT CHANGE*