William schrieb am Samstag, 2. Juli 2022 um 20:14:21 UTC+2:
> On Saturday, July 2, 2022 at 3:02:01 PM UTC-3, WM wrote:
> > William schrieb am Samstag, 2. Juli 2022 um 19:47:26 UTC+2:
> >>. You do not say that something is true for an infinite set.
> > I say that it is true for [each] natural number, none of which is infinite.
>
> Correct. You say that something is true for the elements of |N.

For all elements of ℕ!

> You do not say that something is true for |N

Of course not. ℕ does not appear in
|∩{E(k) : k ∈ ℕ_def}| = ℵ₀
∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
∩{E(k) : k ∈ ℕ} = { }
as an element of ℕ.

Regards, WM

Jim Burns schrieb am Samstag, 2. Juli 2022 um 23:11:28 UTC+2:
> On 7/2/2022 10:49 AM, WM wrote:
> > Gus Gassmann schrieb
> > am Freitag, 1. Juli 2022 um 15:54:35 UTC+2:
> >> On Friday, 1 July 2022 at 09:49:34 UTC-3,
> >> FromTheRafters wrote:
> >>> WM wrote on 7/1/2022 :
>
> >>>> Every k∈|N has ℵo not yet distinguished successors.
> >>>
> >>> Try proving that without using induction.
> >>
> >> Even more to the point, try proving it if n is
> >> a "dark number".
> >
> > Impossible.

> Impossible by design.

Impossible because dark numbers have no discernible order.

> You invent "dark numbers" defined to be
> things

that are required by Cantor's claim of actual infinity and the existence of endsegments implied by it.

|∩{E(k) : k ∈ ℕ_def}| = ℵ₀
∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
∩{E(k) : k ∈ ℕ} = { }

Regards, WM

Jim Burns schrieb am Sonntag, 3. Juli 2022 um 00:10:53 UTC+2:
> On 7/2/2022 12:42 PM, WM wrote:

> > For all definable numbers we obtain
> > |∩{E(k) : k ∈ ℕ_def}| = ℵ₀
> No, you don't obtain that.
>
> That definition is ill-formed.

|∩{E(k) : k <10}| = ℵ₀
|∩{E(k) : k <100}| = ℵ₀
|∩{E(k) : k <10^10}| = ℵ₀
|∩{E(k) : k <10^10^10}| = ℵ₀
and so on for all definable numbers k.

Can you find an exception?

> If we correct your definition to what you say,
> then you obtain |∩{E(k) : k ∈ ℕ_def}| = 0

That is no correction but a misleading change, because I use only such endsegments which satisfy
|∩{E(k) : k ∈ ℕ_def}| = ℵ₀.

> As a consequence,
> each element of ℕ can be reached by
> a terminating process of '+1' steps.

Then try to reach elements with less than ℵ₀ successors.

Regards, WM

On Sunday, July 3, 2022 at 11:08:01 AM UTC-3, WM wrote:
> William schrieb am Samstag, 2. Juli 2022 um 20:14:21 UTC+2:
> > On Saturday, July 2, 2022 at 3:02:01 PM UTC-3, WM wrote:
> > > William schrieb am Samstag, 2. Juli 2022 um 19:47:26 UTC+2:
> > >>. You do not say that something is true for an infinite set.
> > > I say that it is true for [each] natural number, none of which is infinite.
> >
> > Correct. You say that something is true for the elements of |N.
>
> For [each] element of ℕ!

Correct, but not for the set |N.

--
William Hughes

zelos...@gmail.com schrieb am Sonntag, 3. Juli 2022 um 09:15:33 UTC+2:
> lördag 2 juli 2022 kl. 18:37:32 UTC+1 skrev WM:
> > zelos...@gmail.com schrieb am Samstag, 2. Juli 2022 um 19:31:34 UTC+2:
> > > lördag 2 juli 2022 kl. 17:25:15 UTC+1 skrev WM:
> > > >
> > > > For all definable numbers we have the intersection of their endsegments
> > > > |∩{E(k) : k ∈ ℕ_def}| = ℵ₀
> > > false, you keep assuming this but it is false.
> > I defined ℕ_def by this property.
> Nope, you define N_def by N_def={n e N:|∩{E(1), E(2), ..., E(k)}|=ℵ₀}
>
> Which does not result in |∩{E(k) : k ∈ ℕ_def}| = ℵ₀
> It results in |∩{E(k) : k ∈ ℕ_def}| = 0

No, I stop when the intersection is ℵ₀

> > > > The endsegments can only shrink by on number per endsegment
> > > > ∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
> > > >
> > > > What natural numbers reduce the intersection from ℵ₀ to zero?
> > > there is no single specicifc one,

There are numbers which cannot appear as single specific ones. I call them dark numbers.
But we can also say no single specicifc numbers.

Regards, WM

William schrieb am Sonntag, 3. Juli 2022 um 16:23:49 UTC+2:
> On Sunday, July 3, 2022 at 11:08:01 AM UTC-3, WM wrote:
> > William schrieb am Samstag, 2. Juli 2022 um 20:14:21 UTC+2:
> > > On Saturday, July 2, 2022 at 3:02:01 PM UTC-3, WM wrote:
> > > > William schrieb am Samstag, 2. Juli 2022 um 19:47:26 UTC+2:
> > > >>. You do not say that something is true for an infinite set.
> > > > I say that it is true for [each] natural number, none of which is infinite.
> > >
> > > Correct. You say that something is true for the elements of |N.
> >
> > For [each] element of ℕ!
>
> Correct, but not for the set |N.

Of course not. ℕ does not appear in
|∩{E(k) : k ∈ ℕ_def}| = ℵ₀
∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
∩{E(k) : k ∈ ℕ} = { }
as an element of ℕ.

Regards, WM

On Sunday, July 3, 2022 at 11:23:43 AM UTC-3, WM wrote:
> Jim Burns schrieb am Sonntag, 3. Juli 2022 um 00:10:53 UTC+2:

> > each element of ℕ can be reached by
> > a terminating process of '+1' steps.
>
> Then try to reach elements with less than ℵ₀ successors.

This cannot be done. So what? Each *element* of |N has ℵ₀ successors each of which can be reached by a terminating process of '+1' steps. The *set* |N cannot be reached by a terminating process of '+1' steps.

--
William Hughes

William schrieb am Sonntag, 3. Juli 2022 um 16:32:09 UTC+2:
> On Sunday, July 3, 2022 at 11:23:43 AM UTC-3, WM wrote:
> > Jim Burns schrieb am Sonntag, 3. Juli 2022 um 00:10:53 UTC+2:
>
>
> > > each element of ℕ can be reached by
> > > a terminating process of '+1' steps.
> >
> > Then try to reach elements with less than ℵ₀ successors..
> This cannot be done.

That means they remain alway not reached. They are existing only as an undistinguishable dark segments.

> So what? Each *element* of |N has ℵ₀ successors each of which can be reached by a terminating process of '+1' steps.

If each could be reached then not almost all would remain not reached.

>The *set* |N cannot be reached by a terminating process of '+1' steps.

The set is not a number und der discussion.

Regards, WM

On 7/3/2022 9:17 AM, WM wrote:
> Jim Burns schrieb am Samstag, 2. Juli 2022 um 23:11:28 UTC+2:
>> On 7/2/2022 10:49 AM, WM wrote:
>>> Gus Gassmann schrieb
>>> am Freitag, 1. Juli 2022 um 15:54:35 UTC+2:
>>>> On Friday, 1 July 2022 at 09:49:34 UTC-3,
>>>> FromTheRafters wrote:
>>>>> WM wrote on 7/1/2022 :
>>
>>>>>> Every k∈|N has ℵo not yet distinguished successors.
>>>>>
>>>>> Try proving that without using induction.
>>>>
>>>> Even more to the point, try proving it if n is
>>>> a "dark number".
>>>
>>> Impossible.
>
>> Impossible by design.
>
> Impossible because dark numbers have no discernible order.

so they are not numbers.

>
>
>> You invent "dark numbers" defined to be
>> things
>
> that are required by Cantor's claim of actual infinity and the existence of endsegments implied by it.

not at all.

>
> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀

your variable ℕ_def must be finite for that to be true, so correct this equation.

> ∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
> ∩{E(k) : k ∈ ℕ} = { }
>
> Regards, WM
>
>

On 7/3/2022 9:29 AM, WM wrote:
> zelos...@gmail.com schrieb am Sonntag, 3. Juli 2022 um 09:15:33 UTC+2:
>> lördag 2 juli 2022 kl. 18:37:32 UTC+1 skrev WM:
>>> zelos...@gmail.com schrieb am Samstag, 2. Juli 2022 um 19:31:34 UTC+2:
>>>> lördag 2 juli 2022 kl. 17:25:15 UTC+1 skrev WM:
>>>>>
>>>>> For all definable numbers we have the intersection of their endsegments
>>>>> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀
>>>> false, you keep assuming this but it is false.
>>> I defined ℕ_def by this property.
>> Nope, you define N_def by N_def={n e N:|∩{E(1), E(2), ..., E(k)}|=ℵ₀}
>>
>> Which does not result in |∩{E(k) : k ∈ ℕ_def}| = ℵ₀
>> It results in |∩{E(k) : k ∈ ℕ_def}| = 0
>
> No, I stop when the intersection is ℵ₀

no, you have a k there, you stopped at k.

>
>>>>> The endsegments can only shrink by on number per endsegment
>>>>> ∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
>>>>>
>>>>> What natural numbers reduce the intersection from ℵ₀ to zero?
>>>> there is no single specicifc one,
>
> There are numbers which cannot appear as single specific ones. I call them dark numbers.

why have you restricted their appearance ?

> But we can also say no single specicifc numbers.
>
> Regards, WM

söndag 3 juli 2022 kl. 15:08:01 UTC+1 skrev WM:
> William schrieb am Samstag, 2. Juli 2022 um 20:14:21 UTC+2:
> > On Saturday, July 2, 2022 at 3:02:01 PM UTC-3, WM wrote:
> > > William schrieb am Samstag, 2. Juli 2022 um 19:47:26 UTC+2:
> > >>. You do not say that something is true for an infinite set.
> > > I say that it is true for [each] natural number, none of which is infinite.
> >
> > Correct. You say that something is true for the elements of |N.
>
> For all elements of ℕ!
>
> > You do not say that something is true for |N
>
> Of course not. ℕ does not appear in
> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀

this is still false

> ∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
> ∩{E(k) : k ∈ ℕ} = { }
> as an element of ℕ.
>
> Regards, WM

On Sunday, July 3, 2022 at 11:32:04 AM UTC-3, WM wrote:
> William schrieb am Sonntag, 3. Juli 2022 um 16:23:49 UTC+2:
> > On Sunday, July 3, 2022 at 11:08:01 AM UTC-3, WM wrote:
> > > William schrieb am Samstag, 2. Juli 2022 um 20:14:21 UTC+2:
> > > > On Saturday, July 2, 2022 at 3:02:01 PM UTC-3, WM wrote:
> > > > > William schrieb am Samstag, 2. Juli 2022 um 19:47:26 UTC+2:
> > > > >>. You do not say that something is true for an infinite set.
> > > > > I say that it is true for [each] natural number, none of which is infinite.
> > > >
> > > > Correct. You say that something is true for the elements of |N.
> > >
> > > For [each] element of ℕ!
> >
> > Correct, but not for the set |N.

> ∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}

An infinite number of statements, (each statement about an element of |N)

There is no contradiction with the statement
{∩E(k) : k ∈ ℕ} = { }
which is one statement about the set |N.

--
William Hughes

söndag 3 juli 2022 kl. 15:23:43 UTC+1 skrev WM:
> Jim Burns schrieb am Sonntag, 3. Juli 2022 um 00:10:53 UTC+2:
> > On 7/2/2022 12:42 PM, WM wrote:
>
> > > For all definable numbers we obtain
> > > |∩{E(k) : k ∈ ℕ_def}| = ℵ₀
> > No, you don't obtain that.
> >
> > That definition is ill-formed.
>
> |∩{E(k) : k <10}| = ℵ₀
> |∩{E(k) : k <100}| = ℵ₀
> |∩{E(k) : k <10^10}| = ℵ₀
> |∩{E(k) : k <10^10^10}| = ℵ₀
> and so on for all definable numbers k.

no, this goes on for ALL natural numbers k, it does not matter which one.

>
> Can you find an exception?

nope because it applies to ALL natural numbers. "definable" has no definition and is not needed as it is ALL NATURAL NUMBERS!

>
> > If we correct your definition to what you say,
> > then you obtain |∩{E(k) : k ∈ ℕ_def}| = 0

Which is correct because N_def=N

>
> That is no correction but a misleading change, because I use only such endsegments which satisfy
> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀.

Nope, you juse ALL endsegments.

>
> > As a consequence,
> > each element of ℕ can be reached by
> > a terminating process of '+1' steps.
>
> Then try to reach elements with less than ℵ₀ successors.
>
> Regards, WM

On 7/3/2022 9:23 AM, WM wrote:
>
> Jim Burns schrieb am Sonntag, 3. Juli 2022 um 00:10:53 UTC+2:
>> On 7/2/2022 12:42 PM, WM wrote:
>
>>> For all definable numbers we obtain
>>> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀
>> No, you don't obtain that.
>>
>> That definition is ill-formed.
>
> |∩{E(k) : k <10}| = ℵ₀
> |∩{E(k) : k <100}| = ℵ₀
> |∩{E(k) : k <10^10}| = ℵ₀
> |∩{E(k) : k <10^10^10}| = ℵ₀
> and so on for all definable numbers k.
>
> Can you find an exception?
>
>> If we correct your definition to what you say,
>> then you obtain |∩{E(k) : k ∈ ℕ_def}| = 0
>
> That is no correction but a misleading change, because I use only such endsegments which satisfy
> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀.
>
>> As a consequence,
>> each element of ℕ can be reached by
>> a terminating process of '+1' steps.
>
> Then try to reach elements with less than ℵ₀ successors.
>
> Regards, WM

On 7/1/2022 5:02 AM, WM wrote:
> sergi o schrieb am Freitag, 1. Juli 2022 um 03:37:24 UTC+2:
>> WM's "Definition: A natural number is "identified" or (individually) "defined" or "instantiated" if it can be communicated such that sender and receiver
>> understand the same and can link it by a finite initial segment to the origin 0. All other natural numbers are called dark natural numbers."
>>
>>
>>
>>
>> 1. The sender must know the entire "dark" number in order to send it. (The sender cannot make stuff up.)
>>
>> 2. However, all Dark numbers have no digits at all.
>>
>> 3. Therefore there no sender that can send a Dark number anywhere, as there are no number/no digits.
>
> Dark numbers cannot be treated individually. Therefore they cannot be sent.

so you agree with #1, #2 and #3

>>
>> 4. Therefore no Dark Numbers can be ever be "identified" or (individually) "defined" or "instantiated".
>
> The set of defined numbers can be extended by constructing further defined numbers. Then they are no longer dark.

but the sender would have to know all the digits of the dark number to send it, but the number is dark, so he cannot (see #1, #2 and #3 above.)
since he cannot, he cannot send it. (see #5)

>>
>> 5. If the sender knew the digits of the "dark" number before he sends it, the number is known by its digits, therefore is not dark.
>
> Correct.

you agree with #5

>
> Regards, WM
>

On 7/3/2022 9:42 AM, WM wrote:
> William schrieb am Sonntag, 3. Juli 2022 um 16:32:09 UTC+2:
>> On Sunday, July 3, 2022 at 11:23:43 AM UTC-3, WM wrote:
>>> Jim Burns schrieb am Sonntag, 3. Juli 2022 um 00:10:53 UTC+2:
>>
>>
>>>> each element of ℕ can be reached by
>>>> a terminating process of '+1' steps.
>>>
>>> Then try to reach elements with less than ℵ₀ successors.
>> This cannot be done.
>
> That means they remain alway not reached. They are existing only as an undistinguishable dark segments.
>
>> So what? Each *element* of |N has ℵ₀ successors each of which can be reached by a terminating process of '+1' steps.
>
> If each could be reached then not almost all would remain not reached.
>
> >The *set* |N cannot be reached by a terminating process of '+1' steps.
>
> The set is not a number und der discussion.
>
> Regards, WM

On 7/1/2022 5:02 AM, WM wrote:
> sergi o schrieb am Freitag, 1. Juli 2022 um 03:37:24 UTC+2:
>> WM's "Definition: A natural number is "identified" or (individually) "defined" or "instantiated" if it can be communicated such that sender and receiver
>> understand the same and can link it by a finite initial segment to the origin 0. All other natural numbers are called dark natural numbers."
>>
>>
>>
>>
>> 1. The sender must know the entire "dark" number in order to send it. (The sender cannot make stuff up.)
>>
>> 2. However, all Dark numbers have no digits at all.
>>
>> 3. Therefore there no sender that can send a Dark number anywhere, as there are no number/no digits.
>
> Dark numbers cannot be treated individually. Therefore they cannot be sent.

so you agree with #1, #2 and #3

>>
>> 4. Therefore no Dark Numbers can be ever be "identified" or (individually) "defined" or "instantiated".
>
> The set of defined numbers can be extended by constructing further defined numbers. Then they are no longer dark.

but the sender would have to know all the digits of the dark number to send it, but the number is dark, so he cannot (see #1, #2 and #3 above.)
since he cannot, he cannot send it. (see #5)

>>
>> 5. If the sender knew the digits of the "dark" number before he sends it, the number is known by its digits, therefore is not dark.
>
> Correct.

you agree with #5

>
> Regards, WM
>

On Sunday, July 3, 2022 at 11:42:46 AM UTC-3, WM wrote:
> not reached

"reached" is another one of your many many many ways of saying "can be written down". So what if the ℵ₀ successors cannot be written down. Each of the successors has the property that it is the outcome of a terminating process of '+1' steps.

--
William Hughes

William schrieb am Sonntag, 3. Juli 2022 um 16:32:09 UTC+2:
> On Sunday, July 3, 2022 at 11:23:43 AM UTC-3, WM wrote:
> > Jim Burns schrieb am Sonntag, 3. Juli 2022 um 00:10:53 UTC+2:
>
>
> > > each element of ℕ can be reached by
> > > a terminating process of '+1' steps.
> >
> > Then try to reach elements with less than ℵ₀ successors..
> This cannot be done. So what? Each *element* of |N has ℵ₀ successors each of which can be reached by a terminating process of '+1' steps. The *set* |N cannot be reached by a terminating process of '+1' steps..

The reason is that ℕ consists mainly of dark numbers. Note: ℕ is infinite but every reached number belongs to a comparatively tiny finite set.

It is silly to claim that all natnumbers can be defined whereas almost all natnumbers are undefined successors of defined natnumbers.

Regards, WM

zelos...@gmail.com schrieb am Sonntag, 3. Juli 2022 um 16:55:34 UTC+2:
> söndag 3 juli 2022 kl. 15:08:01 UTC+1 skrev WM:
> > William schrieb am Samstag, 2. Juli 2022 um 20:14:21 UTC+2:
> > > On Saturday, July 2, 2022 at 3:02:01 PM UTC-3, WM wrote:
> > > > William schrieb am Samstag, 2. Juli 2022 um 19:47:26 UTC+2:
> > > >>. You do not say that something is true for an infinite set.
> > > > I say that it is true for [each] natural number, none of which is infinite.
> > >
> > > Correct. You say that something is true for the elements of |N.
> >
> > For all elements of ℕ!
> >
> > > You do not say that something is true for |N
> >
> > Of course not. ℕ does not appear in
> > |∩{E(k) : k ∈ ℕ_def}| = ℵ₀
> this is still false

If it was, then you could find a first definable number k with E(k) being finite. You cannot, can you?

Regards, WM

William schrieb am Sonntag, 3. Juli 2022 um 16:55:45 UTC+2:
> On Sunday, July 3, 2022 at 11:32:04 AM UTC-3, WM wrote:

> > > > For [each] element of ℕ!
> > >
> > > Correct, but not for the set |N.
> > ∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k} (#)
> An infinite number of statements, (each statement about an element of |N)

Yes, one statement for every natnumber. There are infinitely many natnumbers. That implies infinitely many statements.
>
> There is no contradiction with the statement
> {∩E(k) : k ∈ ℕ} = { }
> which is one statement about the set |N.

No, it is about the elements of ℕ. And from (#) we find that the empty set has a finite predecessor - or are there dark numbers which do not obey (#). What is it?

Regards, WM

zelos...@gmail.com schrieb am Sonntag, 3. Juli 2022 um 16:57:00 UTC+2:
> söndag 3 juli 2022 kl. 15:23:43 UTC+1 skrev WM:
> > Jim Burns schrieb am Sonntag, 3. Juli 2022 um 00:10:53 UTC+2:
> > > On 7/2/2022 12:42 PM, WM wrote:
> >
> > > > For all definable numbers we obtain
> > > > |∩{E(k) : k ∈ ℕ_def}| = ℵ₀
> > > No, you don't obtain that.
> > >
> > > That definition is ill-formed.
> >
> > |∩{E(k) : k <10}| = ℵ₀
> > |∩{E(k) : k <100}| = ℵ₀
> > |∩{E(k) : k <10^10}| = ℵ₀
> > |∩{E(k) : k <10^10^10}| = ℵ₀
> > and so on for all definable numbers k.
> no, this goes on for ALL natural numbers k, it does not matter which one.

Then this does not go on for all natnumbers?
∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
> >
> > Can you find an exception?
> nope because it applies to ALL natural numbers. "definable" has no definition and is not needed as it is ALL NATURAL NUMBERS!

Then explain how this fails:
∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
> >
> > I use only such endsegments which satisfy
> > |∩{E(k) : k ∈ ℕ_def}| = ℵ₀.

> Nope, you juse ALL endsegments.

why should I? I use only such endsegments which satisfy
|∩{E(k) : k ∈ ℕ_def}| = ℵ₀.

Regards, WM

William schrieb am Sonntag, 3. Juli 2022 um 17:02:07 UTC+2:
> On Sunday, July 3, 2022 at 11:42:46 AM UTC-3, WM wrote:
> > not reached
>
> "reached" is another one of your many many many ways of saying "can be written down".

No.

> So what if the ℵ₀ successors cannot be written down. Each of the successors has the property that it is the outcome of a terminating process of '+1' steps.

No, then all would be used up, none would remain. Compare the enumertaion of all rationals. Will there also remain infinitely many not indexed?

But what is wrong here?
|∩{E(k) : k ∈ ℕ_def}| = ℵ₀
∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
∩{E(k) : k ∈ ℕ} = { }

Regards, WM

On 7/3/2022 10:44 AM, WM wrote:
> William schrieb am Sonntag, 3. Juli 2022 um 17:02:07 UTC+2:
>> On Sunday, July 3, 2022 at 11:42:46 AM UTC-3, WM wrote:
>>> not reached
>>
>> "reached" is another one of your many many many ways of saying "can be written down".
>
> No.
>
>> So what if the ℵ₀ successors cannot be written down. Each of the successors has the property that it is the outcome of a terminating process of '+1' steps.
>
> No, then all would be used up,

you mean the "last one" of an infinite series would be "used up" ?

> Compare the enumertaion of all rationals. Will there also remain infinitely many not indexed?

Liar, all are indexed.

>
> But what is wrong here?
> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀

=> ℕ_def is meaningless

> ∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
> ∩{E(k) : k ∈ ℕ} = { }
>
> Regards, WM

On 7/3/2022 10:29 AM, WM wrote:
> William schrieb am Sonntag, 3. Juli 2022 um 16:32:09 UTC+2:
>> On Sunday, July 3, 2022 at 11:23:43 AM UTC-3, WM wrote:
>>> Jim Burns schrieb am Sonntag, 3. Juli 2022 um 00:10:53 UTC+2:
>>
>>
>>>> each element of ℕ can be reached by
>>>> a terminating process of '+1' steps.
>>>
>>> Then try to reach elements with less than ℵ₀ successors.
>> This cannot be done. So what? Each *element* of |N has ℵ₀ successors each of which can be reached by a terminating process of '+1' steps. The *set* |N cannot be reached by a terminating process of '+1' steps.
>
> The reason is that ℕ consists mainly of dark numbers.

no, there are no dark numbers in ℕ. You have always failed to prove that they exist.

>
> It is silly to claim that all natnumbers can be defined whereas almost all natnumbers are undefined successors of defined natnumbers.

what is sillier is your daffynition of your "defined", with its raps + taps, blinking lights, colors of the rainbow.

>
> Regards, WM

On 7/1/2022 5:02 AM, WM wrote:
> sergi o schrieb am Freitag, 1. Juli 2022 um 03:37:24 UTC+2:
>> WM's "Definition: A natural number is "identified" or (individually) "defined" or "instantiated" if it can be communicated such that sender and receiver
>> understand the same and can link it by a finite initial segment to the origin 0. All other natural numbers are called dark natural numbers."
>>
>>
>>
>>
>> 1. The sender must know the entire "dark" number in order to send it. (The sender cannot make stuff up.)
>>
>> 2. However, all Dark numbers have no digits at all.
>>
>> 3. Therefore there no sender that can send a Dark number anywhere, as there are no number/no digits.
>
> Dark numbers cannot be treated individually. Therefore they cannot be sent.

so you agree with #1, #2 and #3

>>
>> 4. Therefore no Dark Numbers can be ever be "identified" or (individually) "defined" or "instantiated".
>
> The set of defined numbers can be extended by constructing further defined numbers. Then they are no longer dark.

but the sender would have to know all the digits of the dark number to send it, but the number is dark, so he cannot (see #1, #2 and #3 above.)
since he cannot, he cannot send it. (see #5)

>>
>> 5. If the sender knew the digits of the "dark" number before he sends it, the number is known by its digits, therefore is not dark.
>
> Correct.

you agree with #5

>
> Regards, WM

On 7/3/2022 10:40 AM, WM wrote:
> zelos...@gmail.com schrieb am Sonntag, 3. Juli 2022 um 16:57:00 UTC+2:
>> söndag 3 juli 2022 kl. 15:23:43 UTC+1 skrev WM:
>>> Jim Burns schrieb am Sonntag, 3. Juli 2022 um 00:10:53 UTC+2:
>>>> On 7/2/2022 12:42 PM, WM wrote:
>>>
>>>>> For all definable numbers we obtain
>>>>> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀
>>>> No, you don't obtain that.
>>>>
>>>> That definition is ill-formed.
>>>
>>> |∩{E(k) : k <10}| = ℵ₀
>>> |∩{E(k) : k <100}| = ℵ₀
>>> |∩{E(k) : k <10^10}| = ℵ₀
>>> |∩{E(k) : k <10^10^10}| = ℵ₀
>>> and so on for all definable numbers k.
>> no, this goes on for ALL natural numbers k, it does not matter which one.
>
> Then this does not go on for all natnumbers?
> ∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}

that is only the relationship between two adjacent endsegments.

∩(E(1), E(2), ..., E(k+1)) =

E(k+1) = E(k) \ {k}

>>>
>>> Can you find an exception?
>> nope because it applies to ALL natural numbers. "definable" has no definition and is not needed as it is ALL NATURAL NUMBERS!
>
> Then explain how this fails:
> ∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}

which simplifies to E(k+1) = E(k) \ {k}, there is no failure, Deceiver.

>>>
>>> I use only such endsegments which satisfy
>>> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀.
>
>> Nope, you juse ALL endsegments.
>
> why should I? I use only such endsegments which satisfy
> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀.

and ℕ_def is meaningless, Deceiver.

>
> Regards, WM

On 7/3/2022 10:37 AM, WM wrote:
> William schrieb am Sonntag, 3. Juli 2022 um 16:55:45 UTC+2:
>> On Sunday, July 3, 2022 at 11:32:04 AM UTC-3, WM wrote:
>
>>>>> For [each] element of ℕ!
>>>>
>>>> Correct, but not for the set |N.
>>> ∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k} (#)
>> An infinite number of statements, (each statement about an element of |N)
>
> Yes, one statement for every natnumber. There are infinitely many natnumbers. That implies infinitely many statements.
>>
>> There is no contradiction with the statement
>> {∩E(k) : k ∈ ℕ} = { }
>> which is one statement about the set |N.
>
> No, it is about the elements of ℕ.

no, ∩ is an operator on sets E(k)

>
> Regards, WM