William schrieb am Donnerstag, 18. November 2021 um 17:26:47 UTC+1:
> On Thursday, November 18, 2021 at 5:04:49 AM UTC-5, WM wrote:

> > You have to distinguish the natural numbers limited by omega and the natural numbers without limits. I
> Nope. There is only one set of natural numbers, a Peano set. The ordinal omega is greater than any natural number.

But if it exists on the ordinal line, then all natnumbers must be positioned before omega. Hence there must be an end of the linear order.

This is shown by the endsegments. All definable numbers are lost in the sequence of endsegments. But infinitely many numbers remain in all endsegments.

Regards, WM

On Friday, November 19, 2021 at 11:01:59 AM UTC+1, WM wrote:

> Infinitely many elements are in the intersection of all [...] endsegments.

Nonsense. The intersection of all endsegments [each of which is infinite] is empty.

1. The intersection of [the set of] all endsegments is empty.
2. Each and every endsegment is empty.

EOD

Fritz Feldhase schrieb am Freitag, 19. November 2021 um 13:06:14 UTC+1:
> On Friday, November 19, 2021 at 11:01:59 AM UTC+1, WM wrote:
>
> > Infinitely many elements are in the intersection of all [...] endsegments.
>
> Nonsense. The intersection of all endsegments [each of which is infinite] is empty.

You are an enemy of mathematics. In mathemathics the intersection of infinite endsegments is infinite.
>
> 1. The intersection of [the set of] all endsegments is empty.

Yes, but the infinite endsegments maintain infinitely many natural numbers from the first to every infinite endsegment.

> 2. Each and every endsegment is empty.

No. Every definable endsegment is infinite. Bur infinitely many endsegments are exactly aqs many as infinitely many natural numbers. How could more natural numbers exist than endsegments. This would be necessary if all infinitely many endsegments had not lost all natural numbers.

Regards, WM

FromTheRafters schrieb am Freitag, 19. November 2021 um 11:06:17 UTC+1:
> WM expressed precisely :

> > For every given number n the endsegment E(n) is infinite. This does not not
> > change during the whole set of endsegments. Therefore not all numbers are
> > removed. Infinitely many remain in all endsegments.
> >
> > Never thought about this simple contradiction?
> It is not a contradiction, it is just you failing to grasp the ideas
> behind infinite sets.

If these ideas include the superstition of fools, I am glad not to follow them.

How can infinitely many endsegments contain infinitely many natural numbers each? Are there more numbers than endsegments? If not, then the endsegments are emptied as soon as the intersection of endsegments has lost all numbers. May this be a sequential process or a sudden event.

Regards, WM

On Friday, November 19, 2021 at 1:06:14 PM UTC+1, Fritz Feldhase wrote:
> On Friday, November 19, 2021 at 11:01:59 AM UTC+1, WM wrote:
> >
> > Infinitely many elements are in the intersection of all [...] endsegments.
> >
> Nonsense. The intersection of all endsegments [each of which is infinite] is empty.
>
> 1. The intersection of [the set of] all endsegments is empty.

Correction:

> 2. Each and every endsegment is infinite.

EOD

On Friday, November 19, 2021 at 1:47:34 PM UTC+1, WM wrote:
> Fritz Feldhase schrieb am Freitag, 19. November 2021 um 13:06:14 UTC+1:
> >
> > The intersection of all endsegments [each of which is infinite] is empty.

This statement comprises 2 differerent claims:

> > 1. The intersection of [the set of] all endsegments is empty.
> >
> Yes
> >
> > 2. Each and every endsegment is infinite. [<< corrected]

Right:

> Every [...] endsegment is infinite.

Moreover:

[There are as] many endsegments [...] as [...] natural numbers.

Indeed. The set of all endsegments {E(n): n e IN} and the set of all natural numbers IN are equivalent: {E(n): n e IN} ~ IN. Hence card({E(n): n e IN}) = card(IN) = aleph_0.

WM was thinking very hard :
> Fritz Feldhase schrieb am Freitag, 19. November 2021 um 13:06:14 UTC+1:
>> On Friday, November 19, 2021 at 11:01:59 AM UTC+1, WM wrote:
>>
>>> Infinitely many elements are in the intersection of all [...] endsegments.
>>
>> Nonsense. The intersection of all endsegments [each of which is infinite] is
>> empty.
>
> You are an enemy of mathematics. In mathemathics the intersection of

*finitely many* [unneeded fluff "infinite" removed] endsegments is
infinite.

>> 1. The intersection of [the set of] all endsegments is empty.
>
> Yes, but the [edit -infinite-] endsegments maintain infinitely many natural
> numbers from the first to every [edit -infinite-] endsegment.
>
>> 2. Each and every endsegment is empty.
>
> No. Every [edit -definable-] endsegment is infinite. Bur infinitely many
> endsegments are exactly aqs many as infinitely many natural numbers.

Naturals with zero has the same size as naturals without zero. No
endsegment of this set will contain zero as an element, and as such
there are no infinite intersection elements.

Naturals without zero has the same size as naturals without one. No
endsegment of this set will contain one as an element, and as such
there are no infinite intersection elements.

> How could more natural numbers exist than endsegments. This would be
> necessary if all infinitely many endsegments had not lost all natural
> numbers.

You are very confused.

WM wrote on 11/19/2021 :
> FromTheRafters schrieb am Freitag, 19. November 2021 um 11:06:17 UTC+1:
>> WM expressed precisely :
>
>>> For every given number n the endsegment E(n) is infinite. This does not not
>>> change during the whole set of endsegments. Therefore not all numbers are
>>> removed. Infinitely many remain in all endsegments.
>>>
>>> Never thought about this simple contradiction?
>> It is not a contradiction, it is just you failing to grasp the ideas
>> behind infinite sets.
>
> If these ideas include the superstition of fools, I am glad not to follow
> them.

But they don't, so what does that make you?

> How can infinitely many endsegments contain infinitely many natural numbers
> each?

How can two lines both be infinitely long?

> Are there more numbers than endsegments? If not, then the endsegments
> are emptied as soon as the intersection of endsegments has lost all numbers.
> May this be a sequential process or a sudden event.

Stop lying.

On Friday, November 19, 2021 at 7:00:06 AM UTC-5, WM wrote:
> William schrieb am Donnerstag, 18. November 2021 um 17:26:47 UTC+1:
> > On Thursday, November 18, 2021 at 5:04:49 AM UTC-5, WM wrote:
>
> > > You have to distinguish the natural numbers limited by omega and the natural numbers without limits. I
> > Nope. There is only one set of natural numbers, a Peano set. The ordinal omega is greater than any natural number.
> But if it exists on the ordinal line, then all natnumbers must be positioned before omega.
Correct
>Hence there must be an end of the linear order.

Nope, The natural numbers are a Peano set. A Peano set has no "end".

--
William Hughes

On 11/19/2021 6:47 AM, WM wrote:
> Fritz Feldhase schrieb am Freitag, 19. November 2021 um 13:06:14 UTC+1:
>> On Friday, November 19, 2021 at 11:01:59 AM UTC+1, WM wrote:
>>
>>> Infinitely many elements are in the intersection of all [...] endsegments.
>>
>> Nonsense. The intersection of all endsegments [each of which is infinite] is empty.
>
> You are an enemy of mathematics. In mathemathics the intersection of infinite endsegments is infinite.

you are wrong.

we have proved this to you many times.

you are a failure at Math.

>>
>> 1. The intersection of [the set of] all endsegments is empty.
>
> Yes, but the infinite endsegments maintain infinitely many natural numbers from the first to every infinite endsegment.

the intersection of all is not an endsegment.

On 11/19/2021 6:52 AM, WM wrote:
> FromTheRafters schrieb am Freitag, 19. November 2021 um 11:06:17 UTC+1:
>> WM expressed precisely :
>
>>> For every given number n the endsegment E(n) is infinite. This does not not
>>> change during the whole set of endsegments. Therefore not all numbers are
>>> removed. Infinitely many remain in all endsegments.
>>>
>>> Never thought about this simple contradiction?
>> It is not a contradiction, it is just you failing to grasp the ideas
>> behind infinite sets.
>
> If these ideas include the superstition of fools, I am glad not to follow them.

Your ego over-rides learning the fundamentals of Math, and using Proofs.

>
> How can infinitely many endsegments contain infinitely many natural numbers each?

they are defined that way, dummy.

> Are there more numbers than endsegments? If not, then the endsegments are emptied as soon as the intersection of endsegments has lost all numbers. May this be a sequential process or a sudden event.

wrong on all counts, you fail to understand SETS, INTERSECTIONS, NUMBERS, and infinity.

>
> Regards, WM
>

On 11/19/2021 7:52 AM, WM wrote:

> How can infinitely many endsegments contain infinitely
> many natural numbers each?

Something you (WM) haven't done yet is ask yourself
what "infinite" or "finite" mean.

Informally, your answer is because there is no way to step
one-by-one over all the end segments, from one end to the other,

.... and there is no way to step one-by-one over all the
natural numbers in one end segment, any one, from on end
to the other.

No total order of end segments exists such that,
for each split of the end segments, there is a crossing-pair
and that order has a first end segment and a last.

No total order of the natural numbers in any end segment
exists such that, for each split of that end segment,
there is a crossing pair, and that order has a
first natural number and a last.

And this is because, for all the end segments,
there is a total order, the usual order, in which
each end segment E(k) is followed by E(k+1),

....and because, for any one end segment,
there is a total order, the usual order, in which
each natural number k is followed by k+1.

Jim Burns schrieb am Freitag, 19. November 2021 um 22:56:40 UTC+1:
> On 11/19/2021 7:52 AM, WM wrote:
>
> > How can infinitely many endsegments contain infinitely
> > many natural numbers each?
> Something you (WM) haven't done yet is ask yourself
> what "infinite" or "finite" mean.

Infinite here means ℵo, more than any natnumber n.
>
> Informally, your answer is because there is no way to step
> one-by-one over all the end segments, from one end to the other,

But there is a general tool to find numbers that are in endsegments and not in the set of the indices: Try to find a natnumber that belongs to the contents of endsegments but not to the set of their indices. You will fail. That means the set of indices exhausts all natnumbers of ℕ. If all endsegments belonging to this set had additional infinite contents, then a set of ℵo had to succeed a set of ℵo.

> And this is because, for all the end segments,
> there is a total order,

and there is a set of ℵo elements of ℕ used as indices. But every infinite endsegment has ℵo elements of ℕ, which are larger than its index, as its contents. Clash.

Regards, WM

William schrieb am Freitag, 19. November 2021 um 15:05:46 UTC+1:
> On Friday, November 19, 2021 at 7:00:06 AM UTC-5, WM wrote:
> > William schrieb am Donnerstag, 18. November 2021 um 17:26:47 UTC+1:
> > > On Thursday, November 18, 2021 at 5:04:49 AM UTC-5, WM wrote:
> >
> > > > You have to distinguish the natural numbers limited by omega and the natural numbers without limits. I
> > > Nope. There is only one set of natural numbers, a Peano set. The ordinal omega is greater than any natural number.
> > But if it exists on the ordinal line, then all natnumbers must be positioned before omega.
> Correct

That is an end.

> >Hence there must be an end of the linear order.
> Nope, The natural numbers are a Peano set. A Peano set has no "end".
>
omega is an unsurpassable barrier. There the sequence or ordered set ends. That's the difference between a Peano set and a Zermelo-set.

Regards, WM

FromTheRafters schrieb am Freitag, 19. November 2021 um 14:59:22 UTC+1:
> But infinitely many
> > endsegments are exactly as many as infinitely many natural numbers.

Yes. Therefore their contents cannot all be infinite in addition.

Every infinite endsegment has an infinite content that is larger than its index. If all indices have been used up, there is nothing larger possible as content.

Regards, WM

FromTheRafters schrieb am Freitag, 19. November 2021 um 15:04:17 UTC+1:
> WM wrote on 11/19/2021 :

> > How can infinitely many endsegments contain infinitely many natural numbers
> > each?
> How can two lines both be infinitely long?

NThey cannot if the first one starts at zero and the second one is the continuation of the first one.

Regards, WM

On 11/20/2021 12:05 PM, WM wrote:
> FromTheRafters schrieb am Freitag, 19. November 2021 um 14:59:22 UTC+1:
>> But infinitely many
>>> endsegments are exactly as many as infinitely many natural numbers.
>
> Yes. Therefore their contents cannot all be infinite in addition.

there is no addition.

>
> Every infinite endsegment has an infinite content that is larger than its index.

wrong.

> If all indices have been used up, there is nothing larger possible as content.

wrong, there is no "used up", there is no "content" they are called elements,

>
> Regards, WM
>

On 11/20/2021 12:07 PM, WM wrote:
> FromTheRafters schrieb am Freitag, 19. November 2021 um 15:04:17 UTC+1:
>> WM wrote on 11/19/2021 :
>
>>> How can infinitely many endsegments contain infinitely many natural numbers
>>> each?
>> How can two lines both be infinitely long?
>
> NThey cannot if the first one starts at zero and the second one is the continuation of the first one.

Wrong Again.

>
> Regards, WM
>

On Saturday, November 20, 2021 at 1:01:09 PM UTC-5, WM wrote:
> William schrieb am Freitag, 19. November 2021 um 15:05:46 UTC+1:
> > On Friday, November 19, 2021 at 7:00:06 AM UTC-5, WM wrote:
> > > William schrieb am Donnerstag, 18. November 2021 um 17:26:47 UTC+1:
> > > > On Thursday, November 18, 2021 at 5:04:49 AM UTC-5, WM wrote:
> > >
> > > > > You have to distinguish the natural numbers limited by omega and the natural numbers without limits. I
> > > > Nope. There is only one set of natural numbers, a Peano set. The ordinal omega is greater than any natural number.
> > > But if it exists on the ordinal line, then all natnumbers must be positioned before omega.
> > Correct
> That is an end.

Nope, omega is not an element of the Peano set of natural numbers. There is no last natural number.

--
William Hughes

WM explained on 11/20/2021 :
> FromTheRafters schrieb am Freitag, 19. November 2021 um 14:59:22 UTC+1:
>> But infinitely many
>>> endsegments are exactly as many as infinitely many natural numbers.
>
> Yes. Therefore their contents cannot all be infinite in addition.

Why not? Other than just because you say so.

> Every infinite endsegment has an infinite content that is larger than its
> index. If all indices have been used up, there is nothing larger possible as
> content.

Show this.

WM wrote :
> FromTheRafters schrieb am Freitag, 19. November 2021 um 15:04:17 UTC+1:
>> WM wrote on 11/19/2021 :
>
>>> How can infinitely many endsegments contain infinitely many natural numbers
>>> each?
>> How can two lines both be infinitely long?
>
> NThey cannot if the first one starts at zero and the second one is the
> continuation of the first one.

Lines don't start and/or stop.

After serious thinking WM wrote :
> Jim Burns schrieb am Freitag, 19. November 2021 um 22:56:40 UTC+1:
>> On 11/19/2021 7:52 AM, WM wrote:
>>
>>> How can infinitely many endsegments contain infinitely
>>> many natural numbers each?
>> Something you (WM) haven't done yet is ask yourself
>> what "infinite" or "finite" mean.
>
> Infinite here means ℵo, more than any natnumber n.

2^ℵo is also 'larger' than any natural number. Are you saying that ℵo
and 2^ℵo are equal?

On 11/19/2021 2:25 AM, WM wrote:
> Fritz Feldhase schrieb am Donnerstag, 18. November 2021 um 19:28:38 UTC+1:
>> On Wednesday, November 17, 2021 at 12:50:07 PM UTC+1, WM wrote:
>>
>>> |∩{E(k) : k ∈ ℕ}| = ℵ₀ .
>> Nope. Your claim is false.
>>
>> Actually, |∩{E(k) : k ∈ ℕ}| = 0 since ∩{E(k) : k ∈ ℕ} = {}.
>
> This is a contradiction.
> Every infinite endsegment receives its contents from the first one

wrong. you cannot even understand what an endsegment is.

Endsegments are fixed sets, they do not "recieve" nor lose, they do not have "contents" they have elements.

<snip crap>

>
> Regards, WM
>

On Friday, November 19, 2021 at 9:25:16 AM UTC+1, WM wrote:
> Fritz Feldhase schrieb am Donnerstag, 18. November 2021 um 19:28:38 UTC+1:
> > On Wednesday, November 17, 2021 at 12:50:07 PM UTC+1, WM wrote:
> > >
> > > |∩{E(k) : k ∈ ℕ}| = ℵ₀ .
> > >
> > Nope. Your claim is false.
> >
> > Actually, |∩{E(k) : k ∈ ℕ}| = 0 since ∩{E(k) : k ∈ ℕ} = {}.
> >
> This is a contradiction.

Huh?!

You know, for getting a contradiction you need to prove a statement A and its negation ~A.

So what is the statement A you are referring to and where are the proofs for A and ~A (in the context of set theory, say ZFC)?

A predicate with the free variable "k":

> ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀
>
> By the axiom of separation [natural numbers satifying this predicate] can be collected in a set ℕ_def

Indeed, we may define the set

ℕ_def := {k e IN : ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀} .

Now it's easy to show that

ℕ_def = {k e IN : |E(k)| = ℵ₀}

[since An e IN: ∩{E(1), E(2), ..., E(k)} = E(k)]

and finally

ℕ_def = {k : k e IN} = IN

[since An e IN: |E(k)| = ℵ₀].

This PROVES that your claim

> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀ .

is wrong, since

|∩{E(k) : k ∈ ℕ}| = 0

and (see prove above) IN = IN_def.

> ∩{E(k) : k ∈ ℕ} = {}

Exactly. Hence |∩{E(k) : k ∈ ℕ}| = |{}| = 0 .

> <bla bla> only for k [e IN] which have ℵ₀ successors.

Hint: there are no k in IN which DON'T have ℵ₀ successors.

In other words: Each and every k e IN has ℵ₀ successors.

Using symbols:

Ak e IN: card({n e IN : n > k}) = ℵ₀ .

William schrieb am Samstag, 20. November 2021 um 20:16:37 UTC+1:
> On Saturday, November 20, 2021 at 1:01:09 PM UTC-5, WM wrote:

> > > > But if it exists on the ordinal line, then all natnumbers must be positioned before omega.
> > > Correct
> > That is an end.
> Nope, omega is not an element of the Peano set of natural numbers. There is no last natural number.

If there is no last natural number, then there is an empty space before omega. Why?
If there are all natural numbers, then either there is a last one or infinitely many cannot be distinguished; they are dark. This is suggested by the Peano set: Every element of a Paeno set is followed by almost all natural numbers. So these followers cannot be exhausted as individuals. They are not Peano numbers.

Regards, WM