On 11/25/2021 3:03 AM, WM wrote:
> William schrieb am Mittwoch, 24. November 2021 um 17:47:22 UTC+1:
>> On Wednesday, November 24, 2021 at 3:46:36 AM UTC-5, WM wrote:
>> it has no largest element. But every element has almost all ℵo natnumbers beyond itself. This distance cannot be bridged by Peano elements.
>>>> The Peano set of natural numbers does not contain "dark" elements.
>>> It is not actually infinite.
>> Piffle, A Peano set has cardinality aleph_0
>>
> Here you use ℵo to denote a potentially infinite set. Cantor has desinated ℵo to denote actually infinite sets where every element has ℵo successors which can be exhausted collectively in mappings or Cantor lists but cannot be exhausted individually because all individuals defined by Peano cannot and do not exhaust the set.
>
> You understand that there is a difference between exhausting and not exhausting?

sets do not exhausting and not exhausting.

>
> Regards, WM
>

On 11/25/2021 3:49 AM, WM wrote:
> zelos...@gmail.com schrieb am Mittwoch, 24. November 2021 um 13:16:13 UTC+1:
>> onsdag 24 november 2021 kl. 10:25:33 UTC+1 skrev WM:
>
>>> As long as the endsegments are infinite, infinitely many numbers are not lost. If all endsegments are infinite, infinitely many numbers are never lost.
>>>
>> none is ever lost.
>
> n is lost in E(n+1)

wrong. n is not in the set E(n+1)

you do very sloppy bad math

On 11/25/2021 5:05 AM, Fritz Feldhase wrote:
> On Thursday, November 25, 2021 at 11:31:52 AM UTC+1, FromTheRafters asked WM:
>
>> Can't you do *anything* right?
>
> No, he can't.
>

a mistake in every post, like Cracker-Jacks

Serg io wrote:

> On 11/25/2021 5:05 AM, Fritz Feldhase wrote:
>> On Thursday, November 25, 2021 at 11:31:52 AM UTC+1, FromTheRafters
>> asked WM:
>>> Can't you do *anything* right?
>>
>> No, he can't.
>>
> a mistake in every post, like Cracker-Jacks

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On 11/25/2021 3:39 AM, WM wrote:
> Fritz Feldhase schrieb am Mittwoch, 24. November 2021 um 10:48:32 UTC+1:
>> On Wednesday, November 24, 2021 at 9:50:59 AM UTC+1, WM wrote:
>>> zelos...@gmail.com schrieb am Dienstag, 23. November 2021 um 10:55:20 UTC+1:
>>>> EACH and EVERY endsegment has [an infinite intersection] with ANY OTHER endsegment but for all, it is not true.
>>>>
>>> Of course it is true then for all.
>> No, it isn't.
>>> How could you contradict this?
>> By a so called MATHEMATICAL PROOF, you know.
>
> Mathematical proof shows by inclusion monotony: All infinite endsegments have an infinite contents in common. If your "proof" shows the contrary, it is based upon a wrong theory. The contents of any endsegment is remaining there from the first one E(1) = |N. As long as there is contents, the set of non-empty endsegments has a non-empty intersection. You know? Or are you a matheologian? They believe that there is no infinite set dark numbers between every natnumber and ω. But they cannot find any addressable number which is closer to ω. Not to be taken seriously.
>
> Regards, WM
>

too many mistakes to take time to correct.

Each and every sentence has mistakes in them.

Gus Gassmann schrieb am Donnerstag, 25. November 2021 um 13:17:58 UTC+1:
> On Thursday, 25 November 2021 at 05:49:55 UTC-4, WM wrote:
> [...]
> > But infinitely many elements are in all infinite endsegments.
> Ambiguous nonsense, as usual.

You cannot get the meaning of this sentence? Every infinite endsegment has an infinite contents.

Regards, WM

William schrieb am Donnerstag, 25. November 2021 um 14:43:04 UTC+1:
> On Thursday, November 25, 2021 at 4:54:10 AM UTC-5, WM wrote:
> > Fritz Feldhase schrieb am Donnerstag, 25. November 2021 um 10:46:44 UTC+1:
>
> > > THERE ARE NO "potentially infinite sets".
> > >
> > The definable natural numbers are
> the numbers you can write down and are a finite set.

But they are not the same numbers that I can write down tomorrow. They are not the same numbers that you can write down. And when we have written down them, then we can write down further numbers. That is what many call a
"Potentially infinite set"

Regards, WM

FredJeffries schrieb am Donnerstag, 25. November 2021 um 15:53:13 UTC+1:
> On Thursday, November 25, 2021 at 1:54:10 AM UTC-8, WM wrote:
> > Fritz Feldhase schrieb am Donnerstag, 25. November 2021 um 10:46:44 UTC+1:
> > > On Thursday, November 25, 2021 at 10:03:58 AM UTC+1, WM wrote:
> > > > William schrieb am Mittwoch, 24. November 2021 um 17:47:22 UTC+1:
> > > > >
> > > > > A Peano set has cardinality aleph_0
> > > > >
> > > > Here you use ℵo to denote a potentially infinite set.
> > > THERE ARE NO "potentially infinite sets".
> > >
> > The definable natural numbers are a potentially infinite set.
> Your 'definable natural numbers' are/is not a set, as you have elsewhere admitted.

I call them a set in my book "Mathematik für die ersten Semester", 4th ed., De Gruyter, Berlin (2015). In set theory the meaning of set has been restricted to actually infinite sets. There we would use collection instead..

Regards, WM

Crank Wolfgang Mückenheim, aka WM wote:
> FredJeffries schrieb am Donnerstag, 25. November 2021 um 15:53:13 UTC+1:
>> On Thursday, November 25, 2021 at 1:54:10 AM UTC-8, WM wrote:
>>> Fritz Feldhase schrieb am Donnerstag, 25. November 2021 um 10:46:44 UTC+1:
>>>> On Thursday, November 25, 2021 at 10:03:58 AM UTC+1, WM wrote:
>>>>> William schrieb am Mittwoch, 24. November 2021 um 17:47:22 UTC+1:
>>>>>>
>>>>>> A Peano set has cardinality aleph_0
>>>>>>
>>>>> Here you use ℵo to denote a potentially infinite set.
>>>> THERE ARE NO "potentially infinite sets".
>>>>
>>> The definable natural numbers are a potentially infinite set.
>> Your 'definable natural numbers' are/is not a set, as you have elsewhere admitted.
>
> I call them a set in my book "Mathematik für die ersten Semester",

Not a good reference: this book has been written by a crank.

On 11/26/2021 9:57 AM, WM wrote:
> William schrieb am Donnerstag, 25. November 2021 um 14:43:04 UTC+1:
>> On Thursday, November 25, 2021 at 4:54:10 AM UTC-5, WM wrote:
>>> Fritz Feldhase schrieb am Donnerstag, 25. November 2021 um 10:46:44 UTC+1:
>>
>>>> THERE ARE NO "potentially infinite sets".
>>>>
>>> The definable natural numbers are
>> the numbers you can write down and are a finite set.
>
> But they are not the same numbers that I can write down tomorrow.

you do not know that! you haven't written down tomorrows numbers!

>They are not the same numbers that you can write down. And when we have written down them, then we can write down further numbers. That is what many call a
>
> "Potentially infinite set"

liar.

>
> Regards, WM
>

On Friday, November 26, 2021 at 10:57:34 AM UTC-5, WM wrote:
> William schrieb am Donnerstag, 25. November 2021 um 14:43:04 UTC+1:
> > On Thursday, November 25, 2021 at 4:54:10 AM UTC-5, WM wrote:
> > > Fritz Feldhase schrieb am Donnerstag, 25. November 2021 um 10:46:44 UTC+1:
> >
> > > > THERE ARE NO "potentially infinite sets".
> > > >
> > > The definable natural numbers are
> > the numbers you can write down and are a finite set.
> But they are not the same numbers that I can write down tomorrow

Something that changes is not a set. "Potentially infinite set" is nonsense.

--
William Hughes

On 11/26/2021 9:55 AM, WM wrote:
> Gus Gassmann schrieb am Donnerstag, 25. November 2021 um 13:17:58 UTC+1:
>> On Thursday, 25 November 2021 at 05:49:55 UTC-4, WM wrote:
>> [...]
>>> But infinitely many elements are in all infinite endsegments.
>> Ambiguous nonsense, as usual.
>
> You cannot get the meaning of this sentence? Every infinite endsegment has an infinite contents.

your sentence has 2 errors;

all endsegments are infinite by definition

sets have "elements", not "contents"

Try using equations, Math, instead of your off the wall vagueatuities.

>
> Regards, WM
>

William schrieb am Freitag, 26. November 2021 um 17:06:56 UTC+1:
> On Friday, November 26, 2021 at 10:57:34 AM UTC-5, WM wrote:
> > William schrieb am Donnerstag, 25. November 2021 um 14:43:04 UTC+1:
> > > On Thursday, November 25, 2021 at 4:54:10 AM UTC-5, WM wrote:
> > > > Fritz Feldhase schrieb am Donnerstag, 25. November 2021 um 10:46:44 UTC+1:
> > >
> > > > > THERE ARE NO "potentially infinite sets".
> > > > >
> > > > The definable natural numbers are
> > > the numbers you can write down and are a finite set.
> > But they are not the same numbers that I can write down tomorrow
> Something that changes is not a set. "Potentially infinite set" is nonsense.
>
Then call it a collection.

But something that does not change, like the actually infinite set ℕ, has a fixed number of elements. It cannot have same numbers of elements as {0} U ℕ, let alone be in bijection, i.e., in 1-1-correspondence with |Q.

Regards, WM

On Friday, November 26, 2021 at 11:24:10 AM UTC-5, WM wrote:
> William schrieb am Freitag, 26. November 2021 um 17:06:56 UTC+1:
> > On Friday, November 26, 2021 at 10:57:34 AM UTC-5, WM wrote:
> > > William schrieb am Donnerstag, 25. November 2021 um 14:43:04 UTC+1:
> > > > On Thursday, November 25, 2021 at 4:54:10 AM UTC-5, WM wrote:
> > > > > Fritz Feldhase schrieb am Donnerstag, 25. November 2021 um 10:46:44 UTC+1:
> > > >
> > > > > > THERE ARE NO "potentially infinite sets".
> > > > > >
> > > > > The definable natural numbers are
> > > > the numbers you can write down and are a finite set.
> > > But they are not the same numbers that I can write down tomorrow
> > Something that changes is not a set. "Potentially infinite set" is nonsense.
> >
> Then call it a collection.

It is not a set. The natural numbers are a Peano set.

>
Your
> ℕ
is not a Peano set. A Peano set does not contain "dark" elements. The natural numbers are a Peano set.

--
William Hughes

On 11/26/2021 10:03 AM, Python wrote:
> Crank Wolfgang Mückenheim, aka WM wote:
>> FredJeffries schrieb am Donnerstag, 25. November 2021 um 15:53:13 UTC+1:
>>> On Thursday, November 25, 2021 at 1:54:10 AM UTC-8, WM wrote:
>>>> Fritz Feldhase schrieb am Donnerstag, 25. November 2021 um 10:46:44 UTC+1:
>>>>> On Thursday, November 25, 2021 at 10:03:58 AM UTC+1, WM wrote:
>>>>>> William schrieb am Mittwoch, 24. November 2021 um 17:47:22 UTC+1:
>>>>>>>
>>>>>>> A Peano set has cardinality aleph_0
>>>>>>>
>>>>>> Here you use ℵo to denote a potentially infinite set.
>>>>> THERE ARE NO "potentially infinite sets".
>>>>>
>>>> The definable natural numbers are a potentially infinite set.
>>> Your 'definable natural numbers' are/is not a set, as you have elsewhere admitted.
>>
>> I call them a set in my book "Mathematik für die ersten Semester",
>
> Not a good reference: this book has been written by a crank.
>
>

the book should be renamed;

" Trollmathematik für Unwissende "

William schrieb am Freitag, 26. November 2021 um 17:49:58 UTC+1:

> A Peano set does not contain "dark" elements. The natural numbers are a Peano set.
>
Then there is an infinite set of Peano numbers between every natnumber and ω. But you cannot find any Peano number which is closer to ω. Call them unfindable numbers.

Regards, WM

On Saturday, November 27, 2021 at 6:58:05 AM UTC-5, WM wrote:
> William schrieb am Freitag, 26. November 2021 um 17:49:58 UTC+1:
>
> > A Peano set does not contain "dark" elements. The natural numbers are a Peano set.
> >
> Then there is an infinite set of Peano numbers between every natnumber and ω.

Correct. (Note the set of natural numbers and the set of "Peano numbers" is the same thing)

>But you cannot find any Peano number which is closer to ω.

Correct. There is no element of the set of natural numbers which is "closer" to omega.

> Call them unfindable numbers.

Whatever, if anything, "them" are, they are not natural numbers. There is no natural number between the set of natural numbers and omega because there is no last element of the set of natural numbers.

--
William Hughes

William schrieb am Samstag, 27. November 2021 um 16:32:03 UTC+1:
> On Saturday, November 27, 2021 at 6:58:05 AM UTC-5, WM wrote:
> > William schrieb am Freitag, 26. November 2021 um 17:49:58 UTC+1:
> >
> > > A Peano set does not contain "dark" elements. The natural numbers are a Peano set.
> > >
> > Then there is an infinite set of Peano numbers between every natnumber and ω.
> Correct. (Note the set of natural numbers and the set of "Peano numbers" is the same thing)
> >But you cannot find any Peano number which is closer to ω.
> Correct. There is no element of the set of natural numbers which is "closer" to omega.
>
> > Call them unfindable numbers.
>
> Whatever, if anything, "them" are, they are not natural numbers. There is no natural number between the set of natural numbers and omega because there is no last element of the set of natural numbers.

But for all Peano elements we have
∀n ∈ ℕ_P: |ℕ \ {1, 2, 3, ..., n}| = ℵ₀.
There is no way to reduce the remaining set of ℵ₀ to less than ℵ₀ by removing individual natnumbers. Only removing the whole set will succeed. This shows that its elements cannot be removed individually. Are they natural numbers? If there are ℵ₀ natural numbers, then they are of course natural numbers. The Peano numbers are less, because there cannot exist two ℵ₀ sets below ω but the dark set has never less than ℵ₀ elements.

Regards, WM

On 11/28/2021 5:18 AM, WM wrote:
> William schrieb am Samstag, 27. November 2021 um 16:32:03 UTC+1:
>> On Saturday, November 27, 2021 at 6:58:05 AM UTC-5, WM wrote:
>>> William schrieb am Freitag, 26. November 2021 um 17:49:58 UTC+1:
>>>
>>>> A Peano set does not contain "dark" elements. The natural numbers are a Peano set.
>>>>
>>> Then there is an infinite set of Peano numbers between every natnumber and ω.
>> Correct. (Note the set of natural numbers and the set of "Peano numbers" is the same thing)
>>> But you cannot find any Peano number which is closer to ω.
>> Correct. There is no element of the set of natural numbers which is "closer" to omega.
>>
>>> Call them unfindable numbers.
>>
>> Whatever, if anything, "them" are, they are not natural numbers. There is no natural number between the set of natural numbers and omega because there is no last element of the set of natural numbers.
>
> But for all Peano elements we have
> ∀n ∈ ℕ_P: |ℕ \ {1, 2, 3, ..., n}| = ℵ₀.
> There is no way to reduce the remaining set of ℵ₀ to less than ℵ₀ by removing individual natnumbers. Only removing the whole set will succeed.

Wrong. just "remove" an endsegment,

This shows that its elements cannot be removed individually.

wrong. any element can be "removed" at any time.

> Are they natural numbers?

you already stated they are natural numbers.

> If there are ℵ₀ natural numbers, then they are of course natural numbers.

wrong. they were always natural numbers.

>The Peano numbers are less,

Prove it.

> because there cannot exist two ℵ₀ sets below ω but the dark set has never less than ℵ₀ elements.

wrong, and wrong.

>
> Regards, WM
>

WM always posts Wrong Math.

On Sunday, November 28, 2021 at 6:18:12 AM UTC-5, WM wrote:
> . ..for all Peano elements we have
> ∀n ∈ ℕ_P: |ℕ \ {1, 2, 3, ..., n}| = ℵ₀.
> There is no way to reduce the remaining set of ℵ₀ to less than ℵ₀ by removing individual natnumbers.

Since by "removing individual natnumbers" you mean removing a finite set, this is correct. You need to remove a Peano set which has cardinality aleph_0.
However, when you remove the Peano set you do not remove any "dark" elements.

--
William Hughes

fredag 26 november 2021 kl. 17:24:10 UTC+1 skrev WM:
> William schrieb am Freitag, 26. November 2021 um 17:06:56 UTC+1:
> > On Friday, November 26, 2021 at 10:57:34 AM UTC-5, WM wrote:
> > > William schrieb am Donnerstag, 25. November 2021 um 14:43:04 UTC+1:
> > > > On Thursday, November 25, 2021 at 4:54:10 AM UTC-5, WM wrote:
> > > > > Fritz Feldhase schrieb am Donnerstag, 25. November 2021 um 10:46:44 UTC+1:
> > > >
> > > > > > THERE ARE NO "potentially infinite sets".
> > > > > >
> > > > > The definable natural numbers are
> > > > the numbers you can write down and are a finite set.
> > > But they are not the same numbers that I can write down tomorrow
> > Something that changes is not a set. "Potentially infinite set" is nonsense.
> >
> Then call it a collection.
>
> But something that does not change, like the actually infinite set ℕ, has a fixed number of elements. It cannot have same numbers of elements as {0} U ℕ, let alone be in bijection, i.e., in 1-1-correspondence with |Q.
>
> Regards, WM

who says the size of the set changes? it doesn't!

But it can sitll be in bijection with those sets and all be of the same size!

It is a counterintuitive result but very valid!

zelos...@gmail.com schrieb am Montag, 29. November 2021 um 06:51:36 UTC+1:
> fredag 26 november 2021 kl. 17:24:10 UTC+1 skrev WM:
> > William schrieb am Freitag, 26. November 2021 um 17:06:56 UTC+1:
> > It cannot have same numbers of elements as {0} U ℕ, let alone be in bijection, i.e., in 1-1-correspondence with |Q.
> >
> who says the size of the set changes? it doesn't!
>
> But it can sitll be in bijection with those sets and all be of the same size!
>
> It is a counterintuitive result but very valid!

It is a clear contradiction because bijections cause equal *number*. ℕ and {0} U ℕ differ by one number.

Regards, WM

William schrieb am Sonntag, 28. November 2021 um 17:31:25 UTC+1:
> On Sunday, November 28, 2021 at 6:18:12 AM UTC-5, WM wrote:
> > . ..for all Peano elements we have
> > ∀n ∈ ℕ_P: |ℕ \ {1, 2, 3, ..., n}| = ℵ₀.
> > There is no way to reduce the remaining set of ℵ₀ to less than ℵ₀ by removing individual natnumbers.
> Since by "removing individual natnumbers" you mean removing a finite set, this is correct.

It suffices to show that the definable numbers are less than ℵ₀.

> You need to remove a Peano set which has cardinality aleph_0.

The definable numbers are less.
> However, when you remove the Peano set you do not remove any "dark" elements.
>
Individually many can be removed but most (ℵ₀) cannot be removed. That is what I call dark.

Regards, WM

måndag 29 november 2021 kl. 11:28:00 UTC+1 skrev WM:
> zelos...@gmail.com schrieb am Montag, 29. November 2021 um 06:51:36 UTC+1:
> > fredag 26 november 2021 kl. 17:24:10 UTC+1 skrev WM:
> > > William schrieb am Freitag, 26. November 2021 um 17:06:56 UTC+1:
> > > It cannot have same numbers of elements as {0} U ℕ, let alone be in bijection, i.e., in 1-1-correspondence with |Q.
> > >
> > who says the size of the set changes? it doesn't!
> >
> > But it can sitll be in bijection with those sets and all be of the same size!
> >
> > It is a counterintuitive result but very valid!
> It is a clear contradiction because bijections cause equal *number*. ℕ and {0} U ℕ differ by one number.
>
> Regards, WM

Nope, you think that because it is how it is for FINITE sets but that is NOT the case for INFINITE sets.

Infinite sets CAN be set into a bijection with a PROPER subset, DEFINITIONALLY.

That is what defined them and is NOT a contradiction!

WM expressed precisely :
> zelos...@gmail.com schrieb am Montag, 29. November 2021 um 06:51:36 UTC+1:
>> fredag 26 november 2021 kl. 17:24:10 UTC+1 skrev WM:
>>> William schrieb am Freitag, 26. November 2021 um 17:06:56 UTC+1:
>>> It cannot have same numbers of elements as {0} U ℕ, let alone be in
>>> bijection, i.e., in 1-1-correspondence with |Q.
>>>
>> who says the size of the set changes? it doesn't!
>>
>> But it can sitll be in bijection with those sets and all be of the same
>> size!
>>
>> It is a counterintuitive result but very valid!
>
> It is a clear contradiction because bijections cause equal *number*.

With finite sets, yes, more like equal *size* when dealing with
infinite sets.