Gus Gassmann schrieb am Sonntag, 25. Juni 2023 um 22:25:01 UTC+2:
> On Sunday, 25 June 2023 at 13:01:44 UTC-3, WM wrote:

> > > 1. All endsegments are infinite (but)
> > > 2. their intersection is empty.
> > This claim shows utmost stupidity or utmost superstition.
> Disputing this often proved fact

I only ask what makes the endsegments infinite after all visible numbers have been lost.

Regards, WM

Eram semper recta schrieb am Montag, 26. Juni 2023 um 01:23:35 UTC+2:
> On Saturday, 24 June 2023 at 09:08:58 UTC-4, WM wrote:
> > Eram semper recta schrieb am Samstag, 24. Juni 2023 um 14:29:13 UTC+2:
> > > On Friday, 23 June 2023 at 07:41:05 UTC-4, WM wrote:
> > > > We assume that all endsegments E(n) = {n, n+1, n+2, ...} of natural numbers are visible and infinite and have an empty intersection. Then there exists at least one visible endsegment E(k) containing at least one natural number j ≥ k that is not in at least one of its predecessors.
> > > >
> > > > Acceptable?
> > > Yes, your claim is 100% correct.
> > Thank you. It is relieving to see that intelligent persons can understand. I am sure my students will become educated as well.
> You should post these questions and your students' responses here. That would be interesting to see.

I will do so. But my lesson will begin only in October (because in summer I do not teach) and the students will learn all this only by the end of the year.

On the other hand it is clear: Unless all natnumbers have gone, the intersection is not empty. And as long as all endsegments are infinite, infinitely many natnumbers have not gone.

Regards, WM

On Monday, June 26, 2023 at 9:42:14 PM UTC+2, WM wrote:

> what makes all endsegments infinite [?]

Their contents?

Eram semper recta schrieb am Montag, 26. Juni 2023 um 14:19:40 UTC+2:
> On Friday, 23 June 2023 at 07:41:05 UTC-4, WM wrote:
> > We assume that all endsegments E(n) = {n, n+1, n+2, ...} of natural numbers are visible and infinite and have an empty intersection. Then there exists at least one visible endsegment E(k) containing at least one natural number j ≥ k that is not in at least one of its predecessors.
> >
> > Acceptable?
> >
> > Regards, WM
>
> I shared your question with ChatGPT.
>
> WM:
> I have a question about set theory.
>
> Let's assume that all end segments E(n) = {n, n+1, n+2, ...} of natural numbers are visible and infinite and have an empty intersection. Then there exists at least one visible end segment E(k) containing at least one natural number j ≥ k that is not in at least one of its predecessors.
>
> Is this conclusion correct? Please keep your answers directed and avoid adding any extraneous or non-relevant information.
>
> ChatGPT:
> Yes, the conclusion is correct. If all end segments E(n) of natural numbers are visible, infinite, and have an empty intersection, then for any given natural number k, there exists an end segment E(k) that contains at least one natural number j ≥ k that is not present in any of its predecessors.
>
> This can be inferred from the fact that each end segment E(k) contains all natural numbers greater than or equal to k. Therefore, there will always be numbers in E(k) that are not present in any of its predecessors E(n) where n < k.
>
> -----------------------------------------------------------------------------------------------------
>
> This is the opinion of ChatGPT which is not necessarily reliable, but on first inspection, the natural conclusion is that your initial claim is correct.

Thank you. Another piece of circumstantial evidence is the denial of all matheolgians to say what remains in the infinite endsegments when all natural numbers n have been lost in E(n+1).

Regards, WM

Fritz Feldhase schrieb am Montag, 26. Juni 2023 um 17:57:28 UTC+2:

> Therefore, ChatGPT's "argument" is WRONG.

It is wrong that all infinite endsegments have an empty intersection. The argument "n is not in E(n+1)" removes all natnumbers from the sequence of endsegments as well as from their intersection.

Regards, WM

on 6/26/2023, WM supposed :
> Jim Burns schrieb am Sonntag, 25. Juni 2023 um 22:19:18 UTC+2:
>> On 6/25/2023 11:59 AM, WM wrote:
>>> Jim Burns schrieb am Samstag,
>>> 24. Juni 2023 um 18:05:04 UTC+2:
>>>> On 6/24/2023 8:59 AM, WM wrote:
>>
>>>>> It is impossible that
>>>>> all endsegments remain infinite
>>>>> whereas their intersection gets empty.
>>>>> Otherwise you would have a counter example.
>>>>
>>>> ⟨6.02214076×10²³…⟩
>>>
>>> That is not an endsegment.
>> ⟨6.02214076×10²³…⟩ =
>> {i|6.02214076×10²³≤i} ⊆ ℕ
>>>>> Every visible n is lost
>>>
>>>> Each n is visible.
>
> What remains to make all endsegments infinite?
> Invisible.
>
>>> What remains in the infinite endsegments
>>> after all n have been lost?
>> 0 ∈ ⟨0⟩∥⟨1…⟩ ⊇ {common} ∌ 0
>> 0 does not
>> remain in all infinite end segments
>> after all n have been lost.
>
> I did not ask what disappears but what makes all endsegments infinite.

The 'set of' all endsegments is infinite because it is in bijection
with the set of naturals.

WM presented the following explanation :
> Fritz Feldhase schrieb am Sonntag, 25. Juni 2023 um 18:42:43 UTC+2:
>
>> So we have:
>>>> 1. All endsegments are infinite (but)
>>>> 2. their intersection is empty.
>> Finally a true statement:
>
> What is in the infinite endsegments which cause the empty intersection
> because they have lost all natnumbers of E(1)?

They don't have to have lost *all* natural numbers of the previous
endsegment in the sequence, just one being not in the next is enough to
make the intersection an emptyset.

> Note: Those which have not
> lost all natnumbers of E(1) do not cause an empty intersection.

Wrong, again, still.

WM explained :
> Fritz Feldhase schrieb am Sonntag, 25. Juni 2023 um 18:56:25 UTC+2:
>
>> If course "ℵ₀ natnumbers are never mising" is "wrong"
>
> If all endegments are infinite and never empty,

The emptyset is not infinite.

WM wrote :
> FromTheRafters schrieb am Sonntag, 25. Juni 2023 um 18:06:53 UTC+2:
>> It happens that WM formulated :
>>> FromTheRafters schrieb am Samstag, 24. Juni 2023 um 16:15:58 UTC+2:
>>>
>>>> Different infinite sets of natural numbers (endsegments) are not equal
>>>> except in cardinality. Not all symbols are in each set, and as such
>>>> those 'missing' symbols cannot be in the intersection.
>>>
>>> If ℵ₀ natnumbers are never mising, they are in the intersection.
>> Wrong.
>
> What numbers are never missing?

A cardinal number is the size of a set, not some finite natural number
of elements which can be missing or not missing.
>
>
> The 'number' of natural numbers is the size of the set of
>> natural numbers. The number of numbers in the subset of natural numbers
>> {100, 101, 102,...} is the same size, but not the same content.
>
> What is the content of all infinite endsegments which have an empty
> intersection?

The 'set of' all [infinite] endsegments contains all endsegments. Some
of these elements have intersection with others but none have
intersection with *all* of them each and every one at once.

They are all different, but have the same size. Their content differs.

WM formulated the question :
> Fritz Feldhase schrieb am Montag, 26. Juni 2023 um 17:57:28 UTC+2:
>
>> Therefore, ChatGPT's "argument" is WRONG.
>
> It is wrong that all infinite endsegments have an empty intersection. The
> argument "n is not in E(n+1)" removes all natnumbers from the sequence of
> endsegments as well as from their intersection.

That's just crazy talk!

On 6/26/2023 3:42 PM, WM wrote:
> Jim Burns schrieb am Sonntag,
> 25. Juni 2023 um 22:19:18 UTC+2:
>> On 6/25/2023 11:59 AM, WM wrote:
>>> Jim Burns schrieb am Samstag,
>>> 24. Juni 2023 um 18:05:04 UTC+2:
>>>> On 6/24/2023 8:59 AM, WM wrote:

>>>>> It is impossible that
>>>>> all endsegments remain infinite
>>>>> whereas their intersection gets empty.
>>>
>>>>> Otherwise you would have
>>>>> a counter example.
>>>>
>>>> ⟨6.02214076×10²³…⟩
>>>
>>> That is not an endsegment.
>>
>> ⟨6.02214076×10²³…⟩ =
>> {i|6.02214076×10²³≤i} ⊆ ℕ

⟨6.02214076×10²³…⟩ is
an end segment and
a counter-example to your claim.

>>>>> Every visible n is lost
>>>
>>>> Each n is visible.
>
> What remains to make all endsegments
> infinite?

Each element is in non-final.
Each hind segment is 1-ended.
Each hind segment is infinite.

Each element is in some 2-ended fore segment.
Each element is not-in some 1-ended hind segment.
Each element is not common to all hind segments.
Each element is not in the intersection of
all hind segments.

An intersection with no elements in it
is not infinite.

To sum up:
Each hind segment is infinite.
Their intersection is empty.

> Invisible.

Each element is visible.
Each element is non-final.
Each element is not common.

Hind segments infinite.
Intersection empty.

>>> What remains in the infinite endsegments
>>> after all n have been lost?
>>
>> 0 ∈ ⟨0⟩∥⟨1…⟩ ⊇ {common} ∌ 0
>> 0 does not
>> remain in all infinite end segments
>> after all n have been lost.
>
> I did not ask what disappears
> but what makes all endsegments infinite.

You (WM) think that an empty intersection
of all (infinite) end segments
is impossible without an empty end segment.

You are incorrect.

On Monday, 26 June 2023 at 16:43:49 UTC-3, WM wrote:
> Gus Gassmann schrieb am Sonntag, 25. Juni 2023 um 22:25:01 UTC+2:
> > On Sunday, 25 June 2023 at 13:01:44 UTC-3, WM wrote:
>
> > > > 1. All endsegments are infinite (but)
> > > > 2. their intersection is empty.
> > > This claim shows utmost stupidity or utmost superstition.
> > Disputing this often proved fact
> I only ask what makes the endsegments infinite after all visible numbers have been lost.

Every end segment, for instance E(1), will still be infinite, even though it has been used with nfinitely many other infinite end segments to produce an empty intersection. You are *WAY* to stupid to grasp any of this.

On Tuesday, June 27, 2023 at 3:57:41 AM UTC+2, Gus Gassmann wrote:

> Every end segment, for instance E(1), will still be infinite, even though it has been used with infinitely many other infinite end segments to produce an empty intersection.

C'mon, no deterioration at all?

On 6/26/2023 10:08 PM, Fritz Feldhase wrote:
> On Tuesday, June 27, 2023
> at 3:57:41 AM UTC+2, Gus Gassmann wrote:

>> Every end segment, for instance E(1),
>> will still be infinite, even though
>> it has been used with infinitely many
>> other infinite end segments to produce
>> an empty intersection.
>
> C'mon, no deterioration at all?

Well, it should be lubricated
with peano oil.

On Monday, June 26, 2023 at 10:01:55 PM UTC+2, WM wrote:
> Fritz Feldhase schrieb am Montag, 26. Juni 2023 um 17:57:28 UTC+2:
> >
> > Therefore, ChatGPT's "argument" is WRONG.
> >
> It is wrong that all infinite endsegments have an empty intersection.

Nope. It's a trivial mathematical fact.

ChatGTP: "it is true that the intersection of all infinite endsegments defined as E(n) = {n, n+1, n+2, n+3, ...} is the empty set."

On Monday, June 26, 2023 at 9:52:07 PM UTC+2, WM wrote:

> Thank you.

3 questions, 3 answers:

(1) Question: "An endsegement is defined as E(n) = {n, n+1, n+2, n+3, ...}. Is it true that the intersection of all these endsegments is the empty set."

ChatGTP: "Yes, you are correct. The intersection of all the endsegments defined as E(n) = {n, n+1, n+2, n+3, ...} is indeed the empty set.

If we consider [all] of these endsegments, we can see that they have no common elements. Each endsegment starts at a different number, and as we move along the sequence, the numbers only increase. Therefore, there is no number that belongs to all of the endsegments simultaneously.

Since the intersection of sets refers to the elements that are common to all sets being considered, and in this case, there are no common elements, the intersection is empty."

ChatGPT is completely right here!

(2) Question: "An endsegement is defined as E(n) = {n, n+1, n+2, n+3, ...}. Is it true that the intersection of all infinite endsegments is the empty set."

ChatGTP: "Yes, it is true that the intersection of all infinite endsegments defined as E(n) = {n, n+1, n+2, n+3, ...} is the empty set."

If we consider [all] infinite endsegments, they will have no common elements. Each endsegment starts at a different number, and as we move along the sequence, the numbers only increase. Thus, there is no number that belongs to all of the endsegments simultaneously.

Since the intersection of sets refers to the elements that are common to all sets being considered, and in this case, there are no common elements, the intersection is empty."

Once more ChatGPT is completely right here! (Actually, the answer agrees with the answer to question 1., since all endsegments are infinite. Siee question 3.)

(3) Question: "An endsegement is defined as E(n) = {n, n+1, n+2, n+3, ...}. Is it true that all endsegments are infinite "

ChatGTP: "Yes, all endsegments defined as E(n) = {n, n+1, n+2, n+3, ...} are indeed infinite.

By definition, an endsegment extends infinitely in one direction. In this case, each endsegment starts at the number n and continues indefinitely, incrementing by 1 with each subsequent element. Therefore, the set of elements in each endsegment is infinite.

For example, if we take E(1), it would be {1, 2, 3, ...}, which represents the set of all positive integers, extending infinitely in the positive direction. Similarly, E(2) would be {2, 3, 4, ...}, E(3) would be {3, 4, 5, ...}, and so on. In each case, the endsegment continues infinitely without [an] endpoint."

Again, ChatGPT is completely right here!

You are welcome!

On Tuesday, 27 June 2023 at 07:06:07 UTC-3, Fritz Feldhase wrote:
> On Monday, June 26, 2023 at 9:52:07 PM UTC+2, WM wrote:
>
> > Thank you.
>
> 3 questions, 3 answers:
>
>
> (1) Question: "An endsegement is defined as E(n) = {n, n+1, n+2, n+3, ....}. Is it true that the intersection of all these endsegments is the empty set."
>
> ChatGTP: "Yes, you are correct. The intersection of all the endsegments defined as E(n) = {n, n+1, n+2, n+3, ...} is indeed the empty set.
>
> If we consider [all] of these endsegments, we can see that they have no common elements. Each endsegment starts at a different number, and as we move along the sequence, the numbers only increase. Therefore, there is no number that belongs to all of the endsegments simultaneously.
>
> Since the intersection of sets refers to the elements that are common to all sets being considered, and in this case, there are no common elements, the intersection is empty."
>
> ChatGPT is completely right here!
>
>
> (2) Question: "An endsegement is defined as E(n) = {n, n+1, n+2, n+3, ....}. Is it true that the intersection of all infinite endsegments is the empty set."
>
> ChatGTP: "Yes, it is true that the intersection of all infinite endsegments defined as E(n) = {n, n+1, n+2, n+3, ...} is the empty set."
>
> If we consider [all] infinite endsegments, they will have no common elements. Each endsegment starts at a different number, and as we move along the sequence, the numbers only increase. Thus, there is no number that belongs to all of the endsegments simultaneously.
>
> Since the intersection of sets refers to the elements that are common to all sets being considered, and in this case, there are no common elements, the intersection is empty."
>
> Once more ChatGPT is completely right here! (Actually, the answer agrees with the answer to question 1., since all endsegments are infinite. Siee question 3.)
>
>
> (3) Question: "An endsegement is defined as E(n) = {n, n+1, n+2, n+3, ....}. Is it true that all endsegments are infinite "
>
> ChatGTP: "Yes, all endsegments defined as E(n) = {n, n+1, n+2, n+3, ...} are indeed infinite.
>
> By definition, an endsegment extends infinitely in one direction. In this case, each endsegment starts at the number n and continues indefinitely, incrementing by 1 with each subsequent element. Therefore, the set of elements in each endsegment is infinite.
>
> For example, if we take E(1), it would be {1, 2, 3, ...}, which represents the set of all positive integers, extending infinitely in the positive direction. Similarly, E(2) would be {2, 3, 4, ...}, E(3) would be {3, 4, 5, ....}, and so on. In each case, the endsegment continues infinitely without [an] endpoint."
>
> Again, ChatGPT is completely right here!

Can't be!!! ChatGTP learns from absorbing vast amounts of material on the internet. Of course this includes overwhelming amounts of stuff written by mathematicians, and only a small handful of authors such as WM, JG, AP who contradict. Evil mathematicians have gotten to ChatGTP!

On Sunday, June 25, 2023 at 6:01:44 PM UTC+2, WM wrote:
> Fritz Feldhase schrieb am Samstag, 24. Juni 2023 um 19:22:44 UTC+2:
> >
> > 1. All endsegments are infinite (but)
> > 2. their intersection is empty.
> >
> This claim shows utmost stupidity or utmost superstition.

Really?

(1) Question: "An endsegement is defined as E(n) = {n, n+1, n+2, n+3, ...}. Is it true that all endsegments are infinite?"

ChatGTP: "Yes, all endsegments defined as E(n) = {n, n+1, n+2, n+3, ...} are indeed infinite. [...]"

(2) Question: "An endsegement is defined as E(n) = {n, n+1, n+2, n+3, ...}. Is it true that the intersection of all these endsegments is the empty set?"

ChatGTP: "Yes, you are correct. The intersection of all the endsegments defined as E(n) = {n, n+1, n+2, n+3, ...} is indeed the empty set. [...]"

You are welcome!

Fritz Feldhase formulated on Tuesday :
> On Monday, June 26, 2023 at 9:52:07 PM UTC+2, WM wrote:
>
>> Thank you.
>
> 3 questions, 3 answers:
>
>
> (1) Question: "An endsegement is defined as E(n) = {n, n+1, n+2, n+3, ...}.
> Is it true that the intersection of all these endsegments is the empty set."
>
> ChatGTP: "Yes, you are correct. The intersection of all the endsegments
> defined as E(n) = {n, n+1, n+2, n+3, ...} is indeed the empty set.
>
> If we consider [all] of these endsegments, we can see that they have no
> common elements. Each endsegment starts at a different number, and as we move
> along the sequence, the numbers only increase. Therefore, there is no number
> that belongs to all of the endsegments simultaneously.
>
> Since the intersection of sets refers to the elements that are common to all
> sets being considered, and in this case, there are no common elements, the
> intersection is empty."
>
> ChatGPT is completely right here!
>
>
> (2) Question: "An endsegement is defined as E(n) = {n, n+1, n+2, n+3, ...}.
> Is it true that the intersection of all infinite endsegments is the empty
> set."
>
> ChatGTP: "Yes, it is true that the intersection of all infinite endsegments
> defined as E(n) = {n, n+1, n+2, n+3, ...} is the empty set."
>
> If we consider [all] infinite endsegments, they will have no common elements.
> Each endsegment starts at a different number, and as we move along the
> sequence, the numbers only increase. Thus, there is no number that belongs to
> all of the endsegments simultaneously.
>
> Since the intersection of sets refers to the elements that are common to all
> sets being considered, and in this case, there are no common elements, the
> intersection is empty."
>
> Once more ChatGPT is completely right here! (Actually, the answer agrees with
> the answer to question 1., since all endsegments are infinite. Siee question
> 3.)
>
>
> (3) Question: "An endsegement is defined as E(n) = {n, n+1, n+2, n+3, ...}.
> Is it true that all endsegments are infinite "
>
> ChatGTP: "Yes, all endsegments defined as E(n) = {n, n+1, n+2, n+3, ...} are
> indeed infinite.
>
> By definition, an endsegment extends infinitely in one direction. In this
> case, each endsegment starts at the number n and continues indefinitely,
> incrementing by 1 with each subsequent element. Therefore, the set of
> elements in each endsegment is infinite.
>
> For example, if we take E(1), it would be {1, 2, 3, ...}, which represents
> the set of all positive integers, extending infinitely in the positive
> direction. Similarly, E(2) would be {2, 3, 4, ...}, E(3) would be {3, 4, 5,
> ...}, and so on. In each case, the endsegment continues infinitely without
> [an] endpoint."
>
> Again, ChatGPT is completely right here!
>
>
> You are welcome!

I would rather believe you, than it.

Gus Gassmann schrieb am Dienstag, 27. Juni 2023 um 03:57:41 UTC+2:
> On Monday, 26 June 2023 at 16:43:49 UTC-3, WM wrote:
> > Gus Gassmann schrieb am Sonntag, 25. Juni 2023 um 22:25:01 UTC+2:
> > > On Sunday, 25 June 2023 at 13:01:44 UTC-3, WM wrote:
> >
> > > > > 1. All endsegments are infinite (but)
> > > > > 2. their intersection is empty.
> > > > This claim shows utmost stupidity or utmost superstition.
> > > Disputing this often proved fact
> > I only ask what makes the endsegments infinite after all visible numbers have been lost.
> Every end segment, for instance E(1), will still be infinite,

Every endsegment upon which infinitely many endsegments follow will be infinite.

But two consecutive infinite sets in ℕ are impossible. As long as the endsegments are infinite, their number is finite.

Regards, WM

On Wednesday, June 28, 2023 at 7:34:38 PM UTC+2, WM wrote:

> Every endsegment [is] infinite.

And there are infinitely many such endsegments. Even ChatGPT knows that.

> But two consecutive infinite sets in ℕ are impossible.

Die unendliche Menge der Endsegmente enthält aber kein solches Paar von Mengen, Du Trottel.

There are no two natural numbers n, m e IN such that E(n) < E(m) (meaning: Ai e E(n): Aj e E(m): i < j).

> As long as <bla

Quatsch.

On Wednesday, 28 June 2023 at 14:34:38 UTC-3, WM wrote:
[...]

Hogwash, as usual. You are just TOO stupid.

Two consecutive infinite sets in ℕ are impossible.

Regards, WM

Two consecutive infinite sets in ℕ are impossible. As long as the endsegments are infinite, their number is finite.

Regards, WM

On Thursday, 29 June 2023 at 09:45:38 UTC-3, WM wrote:
> Two consecutive infinite sets in ℕ are impossible. As long as the endsegments are infinite, their number is finite.

As I said: Your idiotic ramblings have nothing at all to do with the problem at hand and only show once again that you are TOO stupid. And now FUCK OFF. EOD.