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Between infinite and short there is a big difference. -- G. H. Gonnet

Re: Natural numbers and vases III

Subject	Author
Natural numbers and vases III	WM
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	sergi o
Re: Natural numbers and vases III	sergi o
Re: Natural numbers and vases III	FromTheRafters
Re: Natural numbers and vases III	sergi o
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	sergi o
Re: Natural numbers and vases III	sergi o
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	sergi o
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	sergi o
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	FromTheRafters
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	FromTheRafters
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	FromTheRafters
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	FromTheRafters
Re: Natural numbers and vases III	Surgio
Re: Natural numbers and vases III	Jim Burns
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	Jim Burns
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Surgio
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Surgio
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	Jim Africani
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	FredJeffries
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	William
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Re: Natural numbers and vases III	William
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Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	FredJeffries
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	FredJeffries
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	FromTheRafters
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	Eduardo Faqtardo
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	Archimedes Plutonium
Re: Natural numbers and vases III	Archimedes Plutonium

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Re: Natural numbers and vases III

<7f678c22-010a-18ba-300e-6ba18fcc136d@att.net>

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Tue, 30 Aug 2022 15:45:27 -0400
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by: Jim Burns - Tue, 30 Aug 2022 19:45 UTC

On 8/30/2022 3:13 PM, WM wrote:

> Inclusion monotony shows that
> all those endsegments have one and the same
> infinite set of natnumbers in common.

A set of two end segments has
a last (⊇) end segment.

The last end segment is their intersection.

No end segment is last (⊇) of all
end segments.

No end segment is the intersection of
all end segments.

All end segments are infinite and
the intersection of all end segments
is empty.

End segments and intersection are different.
So, this is not a contradiction.

Re: Natural numbers and vases III

<telpfc$anj$1@gioia.aioe.org>

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From: inva...@invalid.com (Sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Tue, 30 Aug 2022 14:46:51 -0500
Organization: Aioe.org NNTP Server
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by: Sergio - Tue, 30 Aug 2022 19:46 UTC

On 8/30/2022 2:25 PM, WM wrote:
> Jim Burns schrieb am Dienstag, 30. August 2022 um 20:43:01 UTC+2:
>> On 8/30/2022 10:03 AM, WM wrote:
>>> Jim Burns schrieb am Montag,
>>> 29. August 2022 um 16:48:13 UTC+2:
>>
>>>> Very closely related to the set of all FISONs,
>>>> there is the set of all FISON-ends.
>>>
>>> ℕ_def
>>
>> ⋃𝓕
>> If a FISON exists which n ends, then n ∈ ⋃𝓕
>> If n ∈ ⋃𝓕 then a FISON exists which n ends.
>>
>> Whatever you call the set of all FISON-ends,
>> it remains the set of all FISON-ends.
>
> But it is not a set.

wrong, he said it is a set.

>>
>> The rest of us call it ℕ not ℕ_def
>> Changing its name does not change _it_
>
> But the rest of you believe that the intersection of infinite endsegments could be empty.

wrong, the intersection of all endsegments is empty.

>
> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀ .

wrong. Math proves you are wrong.

assume k is in the intersection of all endsegments. However k is not in E(k+1). therefore, k is not in the intersection.

>>
>>>> However,
>>>> unlike a FISON, the set of all FISON-ends
>>>> does not end anywhere.
>>>
>>> With n also n^n^n^n is an element.
>>
>> With n also ⌈(((8n+1)¹ᐟ²+1)/2)⌉ is an element.
>
> Irrelevant. All smaller numbers than n are contained in ℕ_def.

silly, discard your silly _def confusinator

>
> Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Tue, 30 Aug 2022 19:51 UTC

Gus Gassmann schrieb am Dienstag, 30. August 2022 um 19:48:23 UTC+2:
> On Tuesday, 30 August 2022 at 11:59:45 UTC-3, WM wrote:
> [...]
> > > There is no such endsegment. So what? There is no endsegment that has an element in common with all its successors.
> > Wrong. Then a successor would exist without an element in common with some predecessor.
> For every end segment E(n) and every end segment E(m) there exists a natural number k in E(n) \intersect E(m).

Not only one, but infinitely many.

> There does *NOT* exist a natural number k such that for every E(n) and E(m) [k in E(n) \intersect E(m)].

What obstacle would hinder to collect as many definable endsegments as possible? Would their intersection be empty?
>
> Quit showing off your skills at swapping quantifiers

Collecting as many definable endsegments as possible means swapping? Why?

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Tue, 30 Aug 2022 19:58 UTC

Jim Burns schrieb am Dienstag, 30. August 2022 um 21:45:36 UTC+2:
> On 8/30/2022 3:13 PM, WM wrote:
>
> > Inclusion monotony shows that
> > all those endsegments have one and the same
> > infinite set of natnumbers in common.
> A set of two end segments has
> a last (⊇) end segment.

All sets of two infinite endsegments do not have a last endsegment. Nevertheless their intersection is infinite.
>
> The last end segment is their intersection.
>
> No end segment is last (⊇) of all
> end segments.
>
> No end segment is the intersection of
> all end segments.

Nevertheless the intersection of all definable endsegments is infinite. It is infinite by definition because the set of definable endsegments is defined by this condition. Because I say so!
>
> All end segments are infinite and
> the intersection of all end segments
> is empty.

Because you say so? As long as infinite endsegments are concerned, they have infinitely many elements in common with all their predecessors. Hence an infinite intersection:
∀k ∈ ℕ: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀
>
> End segments and intersection are different.

Not for definable endsegments.
> So, this is not a contradiction.

For definable endsegments both are identical. Hence there is a contradiction.

Regards, WM.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: wpihug...@gmail.com (William)
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by: William - Tue, 30 Aug 2022 20:05 UTC

On Tuesday, August 30, 2022 at 4:13:40 PM UTC-3, WM wrote:
> William schrieb am Dienstag, 30. August 2022 um 20:13:26 UTC+2:
> > On Tuesday, August 30, 2022 at 2:42:23 PM UTC-3, WM wrote:
> > > William schrieb am Dienstag, 30. August 2022 um 17:30:01 UTC+2:
>
> > > Show an infinite endsegment which has not infinitely many elements together with all infinite endsegments, predecessors as well as successors. If you can't, and it is clear that you can't,
> > So what? Each pair of endsegements (A,B), has infinite set S(A,B) of elements in common. However, there is no single unchanging set S that works for all pairs A,B.
> That is irrelevant. If every infinite endsegment E_m has infinitely many natnumbers in common with every infinite endsegment E(n),then there is no endsegment, which has only a finite number of endsegments in common with any infinite endsegment.

Correct. Note the infinite set can and does change.

>Inclusion monotony shows that all those endsegments have one and the same infinite set of natnumbers in common.,

Nope. It shows no such thing.

Re: Natural numbers and vases III

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Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Tue, 30 Aug 2022 15:08:36 -0500
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by: Sergio - Tue, 30 Aug 2022 20:08 UTC

On 8/30/2022 2:58 PM, WM wrote:
>
> Jim Burns schrieb am Dienstag, 30. August 2022 um 21:45:36 UTC+2:
>> On 8/30/2022 3:13 PM, WM wrote:
>>
>>> Inclusion monotony shows that
>>> all those endsegments have one and the same
>>> infinite set of natnumbers in common.
>> A set of two end segments has
>> a last (⊇) end segment.
>
> All sets of two infinite endsegments do not have a last endsegment. Nevertheless their intersection is infinite.
>>
>> The last end segment is their intersection.
>>
>> No end segment is last (⊇) of all
>> end segments.
>>
>> No end segment is the intersection of
>> all end segments.
>
> Nevertheless the intersection of all definable endsegments is infinite.

meaningless.

> It is infinite by definition because the set of definable endsegments is defined by this condition. Because I say so!

wrong, and please use ALL CAPS on your "Because I say so!" looneytuneary.

>>
>> All end segments are infinite and
>> the intersection of all end segments
>> is empty.
>
> Because you say so? As long as infinite endsegments are concerned, they have infinitely many elements in common with all their predecessors.

Wrong. E(1) does not have any in common.

> Hence an infinite intersection:
> ∀k ∈ ℕ: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀
>>
>> End segments and intersection are different.
>
> Not for definable endsegments.

your diversionator word "definable" is QuaCkeRy. It signals BAD MATH AHEAD.

>
>> So, this is not a contradiction.
>
> For definable endsegments both are identical. Hence there is a contradiction.

Wrong, your diversonary word "definable" is unmasked, your balloon has been popped !

you remain deflated, no math from you...

>
> Regards, WM.

Re: Natural numbers and vases III

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Tue, 30 Aug 2022 16:09:14 -0400
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by: Jim Burns - Tue, 30 Aug 2022 20:09 UTC

On 8/30/2022 3:25 PM, WM wrote:
> Jim Burns schrieb am Dienstag,
> 30. August 2022 um 20:43:01 UTC+2:
>> On 8/30/2022 10:03 AM, WM wrote:
>>> Jim Burns schrieb am Montag,
>>> 29. August 2022 um 16:48:13 UTC+2:

>>>> Very closely related to the set of all FISONs,
>>>> there is the set of all FISON-ends.
>>>
>>> ℕ_def
>>
>> ⋃𝓕
>> If a FISON exists which n ends, then n ∈ ⋃𝓕
>> If n ∈ ⋃𝓕 then a FISON exists which n ends.
>>
>> Whatever you call the set of all FISON-ends,
>> it remains the set of all FISON-ends.
>
> But it is not a set.

<WM<FromTheRafters>>
>>
>> Then what is a set?
>
> As I said:
> All elements are there,
> none can be added.
>
</WM<FromTheRafters>>

All the FISON-ends are in the collection of
FISON-ends. None can be added.

Also,
everything in the collection of FISON-ends
is a a FISON-end. None can be removed.

----
>>>> However,
>>>> unlike a FISON, the set of all FISON-ends
>>>> does not end anywhere.
>>>
>>> With n also n^n^n^n is an element.
>>
>> With n also ⌈(((8n+1)¹ᐟ²+1)/2)⌉ is an element.
>
> Irrelevant.
> All smaller numbers than n are contained in ℕ_def.

Each fraction p/q has its own index n
n = (p+q-1)(p+q-2)/2+p

We know that fractions do not share indexes
because we know (I can show you)
p+q = ⌈(((8n+1)¹ᐟ²+1)/2)⌉

Re: Natural numbers and vases III

<telr2r$sng$2@gioia.aioe.org>

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From: inva...@invalid.com (Sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Tue, 30 Aug 2022 15:14:18 -0500
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by: Sergio - Tue, 30 Aug 2022 20:14 UTC

On 8/30/2022 2:36 PM, WM wrote:
> Jim Burns schrieb am Dienstag, 30. August 2022 um 21:20:45 UTC+2:
>> On 8/30/2022 10:47 AM, WM wrote:
>>> Jim Burns schrieb am Montag,
>>> 29. August 2022 um 20:28:24 UTC+2:
>>
>>>> but that dark numbers don't appear in
>>>> any of the "paradoxes" that WM presents.
>>>
>>> ∩{E(n) | n ∈ ℕ} = { }.
>> ∩{E(n) | n ∈ ⋃𝓕} = { }
>

>> "Because I (WM) say so"
>> is not an argument.
>

>
> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀ .

wrong, your bullshit term "defined" or "_def" is your confusinator, and it says BAD MATH AHEAD.

>
> Regards, WM

Re: Natural numbers and vases III

<5e6af30c-8a96-b544-c372-be7cf70e9416@att.net>

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Tue, 30 Aug 2022 16:30:20 -0400
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by: Jim Burns - Tue, 30 Aug 2022 20:30 UTC

On 8/30/2022 3:51 PM, WM wrote:

> What obstacle would hinder to collect
> as many definable endsegments as possible?

No obstacle. Say
"This collection contains all and only
(definable) end segments."

Nothing can be added.
Nothing can be removed.

> Would their intersection be empty?

Yes, empty.

Do you see that this is a different question?

Each collection _up to_ an end segment
has its intersection = that end segment.

No collection _up to_ an end segment
is the collection of all end segments.

>> Quit showing off your skills at
>> swapping quantifiers
>
> Collecting as many definable endsegments
> as possible means swapping? Why?

It is an invalid swap to step from
| For each natural k,
| there is an infinite set E,
| all of E is after k

to
| There is an infinite set E,
| for each natural k,
| all of E is after k

It is an invalid swap to step from
| For each column,
| there is a row,
| where they meet is white.

to
| There is a row,
| for each column,
| where they meet is white.

Re: Natural numbers and vases III

<tem2jm$1j216$1@dont-email.me>

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From: FTR...@nomail.afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Tue, 30 Aug 2022 18:22:42 -0400
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by: FromTheRafters - Tue, 30 Aug 2022 22:22 UTC

William pretended :
> On Tuesday, August 30, 2022 at 4:13:40 PM UTC-3, WM wrote:
>> William schrieb am Dienstag, 30. August 2022 um 20:13:26 UTC+2:
>>> On Tuesday, August 30, 2022 at 2:42:23 PM UTC-3, WM wrote:
>>>> William schrieb am Dienstag, 30. August 2022 um 17:30:01 UTC+2:
>>>> Show an infinite endsegment which has not infinitely many elements
>>>> together with all infinite endsegments, predecessors as well as
>>>> successors. If you can't, and it is clear that you can't,
>>> So what? Each pair of endsegements (A,B), has infinite set S(A,B) of
>>> elements in common. However, there is no single unchanging set S that works
>>> for all pairs A,B.
>> That is irrelevant. If every infinite endsegment E_m has infinitely many
>> natnumbers in common with every infinite endsegment E(n),then there is no
>> endsegment, which has only a finite number of endsegments in common with any
>> infinite endsegment.
>
> Correct. Note the infinite set can and does change.

I disagree, but there are many different infinite sets.

Re: Natural numbers and vases III

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Tue, 30 Aug 2022 19:44:50 -0400
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by: Jim Burns - Tue, 30 Aug 2022 23:44 UTC

On 8/30/2022 3:58 PM, WM wrote:
> Jim Burns schrieb am Dienstag,
> 30. August 2022 um 21:45:36 UTC+2:
>> On 8/30/2022 3:13 PM, WM wrote:

>>> Inclusion monotony shows that
>>> all those endsegments have one and the same
>>> infinite set of natnumbers in common.
>>
>> A set of two end segments has
>> a last (⊇) end segment.
>
> All sets of two infinite endsegments
> do not have a last endsegment.

No.
For each set {E',E"} of (infinite) end segments,
E' ⊆ E" or E" ⊆ E'

| E' ⊆ E" or E' ⊈ E"
| | (i)
| E' ⊆ E"
| Done.
| | (ii)
| E' ⊈ E"
| | Some m exists such that
| m ∈ E' & m ∉ E"
| || Assume j ∈ E"
|| E" is an end segment.
|| If j < m then j ∉ E"
|| m < j
||
|| E' is an end segment.
|| m ∈ E'
|| j ∈ E'
| | If j ∈ E" then j ∈ E'
| E" ⊆ E'
| Done.

Therefore,
E' ⊆ E" or E" ⊆ E'

> All sets of two infinite endsegments
> do not have a last endsegment.

No.
All sets of two (infinite) end segments
have a last end segment.

Re: Natural numbers and vases III

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by: Archimedes Plutonium - Wed, 31 Aug 2022 00:48 UTC

Move sickfuck Wolfgang Mueckenheim to sci.logic along with his long list of repliers, for these sickfucks cannot notice mistakes of math nor ever the mistakes. Intellectual shitheads like WM only clutter ups sci.math with his mindless "dark numbers and endless prattle on Cantor" can only confuse young readers.

Ruth Charney,Andrew Wiles, Terence Tao, Thomas Hales, John Stillwell, Jill Pipher, Ruth Charney, Ken Ribet, Andrew Beal, John Baez, Roger Penrose, Gerald Edgar, AMS, no-one there can do a Geometry Proof of Fundamental Theorem of Calculus, all they can offer is a limit analysis, so shoddy in logic they never realized that "analyzing" is not the same as "proving" for analyzing is much in the same as "measuring but not proving". And yet, none can do a geometry proof and the reason is quite clear for none can even see that the slant cut in single right-circular cone is a Oval, never the ellipse. So they could never do a geometry proof of FTC even if they wanted to. For they have no logical geometry brain to begin to do anything geometrical. Is it that Andrew Wiles and Terence Tao cannot understand the slant cut in single cone is an Oval, never the ellipse, or is it the foolish Boole logic they teach of 2 OR 1 = 3 with AND as subtraction? Not having a Logical brain to do math, for any rational person would be upset by Wiles, Tao saying truth table of AND is TFFF when it actually is TTTF. Is that why neither Terence Tao or Andrew Wiles can do a geometry proof Fundamental Theorem of Calculus?

Maybe they need to take up Earle Jones offer to wash dishes or pots at Stanford Univ or where ever, for they sure cannot do mathematics.
Why are these people failures of Math?? For none can even contemplate these 4 questions.

1) think a slant cut in single cone is a ellipse when it is proven to be a Oval, never the ellipse. For the cone and oval have 1 axis of symmetry, while ellipse has 2.
2) think Boole logic is correct with AND truth table being TFFF when it really is TTTF in order to avoid 2 OR 1 =3 with AND as subtraction
3) can never do a geometry proof of Fundamental Theorem of Calculus and are too ignorant in math to understand that analysis of something is not proving something in their "limit hornswaggle"
4) too stupid in science to ask the question of physics-- is the 1897 Thomson discovery of a 0.5MeV particle actually the Dirac magnetic monopole and that the muon is the true electron of atoms stuck inside a 840MeV proton torus doing the Faraday law. Showing that Peter Higgs, Sheldon Glashow, Ed Witten, John Baez, Roger Penrose, Arthur B. McDonald are sapheads when it comes to logical thinking in physics with their do nothing proton, do nothing electron.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: horand.g...@gmail.com (Gus Gassmann)
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by: Gus Gassmann - Wed, 31 Aug 2022 01:58 UTC

On Tuesday, 30 August 2022 at 16:51:24 UTC-3, WM wrote:
> Gus Gassmann schrieb am Dienstag, 30. August 2022 um 19:48:23 UTC+2:
> > On Tuesday, 30 August 2022 at 11:59:45 UTC-3, WM wrote:
> > [...]
> > > > There is no such endsegment. So what? There is no endsegment that has an element in common with all its successors.
> > > Wrong. Then a successor would exist without an element in common with some predecessor.
> > For every end segment E(n) and every end segment E(m) there exists a natural number k in E(n) \intersect E(m).
> Not only one, but infinitely many.

One is enough.

> > There does *NOT* exist a natural number k such that for every E(n) and E(m) [k in E(n) \intersect E(m)].
> What obstacle would hinder to collect as many definable endsegments as possible? Would their intersection be empty?

The fact that no end segment E(n+1) contains the minimum element n associated with E(n) makes an infinite intersection of (distinct) end segments the empty set. Why do you bring your SHIT up over and over again? You are a FUCKING MORON, and a pompous, narcissistic ASSHOLE to boot.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: zelos.ma...@gmail.com (zelos...@gmail.com)
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by: zelos...@gmail.com - Wed, 31 Aug 2022 05:05 UTC

tisdag 30 augusti 2022 kl. 15:42:23 UTC+2 skrev WM:
> Fritz Feldhase schrieb am Montag, 29. August 2022 um 16:07:22 UTC+2:
> > On Monday, August 29, 2022 at 3:27:52 PM UTC+2, WM wrote:
> >
> > > you are not aware of the fact that dark numbers exist and that your proofs are invalid for dark numbers.
> > Yeah, I have to admit that I'm indeed not aware of these "facts".
> But you need them to understand set theory correctly.

We do not need "dark numbers" to understand them as they do not exist.

> If no dark numbers would blow up the set ℕ to accomplish

"blow up" is not a mathematical term.

> ∩{E(k) : k ∈ ℕ} = { }
> then only the elements of
> ∀k ∈ ℕ: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀
> would exist and therefore
> ∩{E(k) : k ∈ ℕ} =/= { } .
>
> Regards, WM

Non-sequiter galore.

Re: Natural numbers and vases III

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From: FTR...@nomail.afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Wed, 31 Aug 2022 06:40:48 -0400
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by: FromTheRafters - Wed, 31 Aug 2022 10:40 UTC

zelos...@gmail.com explained on 8/31/2022 :
> tisdag 30 augusti 2022 kl. 15:42:23 UTC+2 skrev WM:
>> Fritz Feldhase schrieb am Montag, 29. August 2022 um 16:07:22 UTC+2:
>>> On Monday, August 29, 2022 at 3:27:52 PM UTC+2, WM wrote:
>>>
>>>> you are not aware of the fact that dark numbers exist and that your proofs
>>>> are invalid for dark numbers.
>>> Yeah, I have to admit that I'm indeed not aware of these "facts".
>> But you need them to understand set theory correctly.
>
> We do not need "dark numbers" to understand them as they do not exist.
>
>> If no dark numbers would blow up the set ℕ to accomplish
>
> "blow up" is not a mathematical term.

Not so fast :)

https://en.wikipedia.org/wiki/Blowing_up

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Wed, 31 Aug 2022 12:34 UTC

Gus Gassmann schrieb am Mittwoch, 31. August 2022 um 03:58:45 UTC+2:
> On Tuesday, 30 August 2022 at 16:51:24 UTC-3, WM wrote:
> > Gus Gassmann schrieb am Dienstag, 30. August 2022 um 19:48:23 UTC+2:
> > > On Tuesday, 30 August 2022 at 11:59:45 UTC-3, WM wrote:
> > > [...]
> > > > > There is no such endsegment. So what? There is no endsegment that has an element in common with all its successors.
> > > > Wrong. Then a successor would exist without an element in common with some predecessor.
> > > For every end segment E(n) and every end segment E(m) there exists a natural number k in E(n) \intersect E(m).
> > Not only one, but infinitely many.
> One is enough.

One alone would be too much. It was identifyable.

> > > There does *NOT* exist a natural number k such that for every E(n) and E(m) [k in E(n) \intersect E(m)].
> > What obstacle would hinder to collect as many definable endsegments as possible? Would their intersection be empty?

> The fact that no end segment E(n+1) contains the minimum element n associated with E(n) makes an infinite intersection of (distinct) end segments the empty set.

It makes the sequence of endsegments get empty too.

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Wed, 31 Aug 2022 12:41 UTC

William schrieb am Dienstag, 30. August 2022 um 22:05:23 UTC+2:
> On Tuesday, August 30, 2022 at 4:13:40 PM UTC-3, WM wrote:
> > William schrieb am Dienstag, 30. August 2022 um 20:13:26 UTC+2:
> > > On Tuesday, August 30, 2022 at 2:42:23 PM UTC-3, WM wrote:
> > > > William schrieb am Dienstag, 30. August 2022 um 17:30:01 UTC+2:
> >
> > > > Show an infinite endsegment which has not infinitely many elements together with all infinite endsegments, predecessors as well as successors. If you can't, and it is clear that you can't,
> > > So what? Each pair of endsegements (A,B), has infinite set S(A,B) of elements in common. However, there is no single unchanging set S that works for all pairs A,B.
> > That is irrelevant. If every infinite endsegment E_m has infinitely many natnumbers in common with every infinite endsegment E(n),then there is no endsegment, which has only a finite number of endsegments in common with any infinite endsegment.
> Correct. Note the infinite set can and does change.

It decreases with increasing index but remains infinite.

> >Inclusion monotony shows that all those endsegments have one and the same infinite set of natnumbers in common.,
> Nope. It shows no such thing.

An infinite set which decreases maintains infinitely many numbers.

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Wed, 31 Aug 2022 12:47 UTC

Jim Burns schrieb am Dienstag, 30. August 2022 um 22:09:23 UTC+2:
> On 8/30/2022 3:25 PM, WM wrote:
> > Jim Burns schrieb am Dienstag,
> > 30. August 2022 um 20:43:01 UTC+2:

> >> Whatever you call the set of all FISON-ends,
> >> it remains the set of all FISON-ends.
> >
> > But it is not a set.
> <WM<FromTheRafters>>
> >>
> >> Then what is a set?
> >
> > As I said:
> > All elements are there,
> > none can be added.

And that is not so for the collection of FISONs.

> All the FISON-ends are in the collection of
> FISON-ends. None can be added.

The FISONs are counted by the content of FISONs. The content is never infinite. So there is a last one. Take it, I will show you a larger one.

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Wed, 31 Aug 2022 13:02 UTC

Jim Burns schrieb am Dienstag, 30. August 2022 um 22:30:29 UTC+2:
> On 8/30/2022 3:51 PM, WM wrote:
>
> > What obstacle would hinder to collect
> > as many definable endsegments as possible?
> No obstacle. Say
> "This collection contains all and only
> (definable) end segments."

and has an infinite intersection:
|∩{E(1), E(2), ..., E(k)}| = ℵ₀
>
> Nothing can be added.
> Nothing can be removed.

That is wrong.

> > Would their intersection be empty?
> Yes, empty.
>
> Do you see that this is a different question?

No, it is not a question because there is a definition: Collect all endsegments which have an infinite intersection. Your answer to my qustion shows that you would deny everything contradicting set theory, even if ist is contradicting simülest logic.
> .

> Each collection _up to_ an end segment
> has its intersection = that end segment.

Right.
>
> No collection _up to_ an end segment
> is the collection of all end segments.

Every definable endsegment can act as the last one. An endsegment that cannot act as a last one cannot be definable.

I do not say for each k. I use only those endsegments which show an infinite intersection together. That is not for all k, but for all definable k, i.e., as long as the intersection is infinity.

There is no swap.
Please correct your misunderstanding or confess that you do not accept:
For those k for which the intersection is infinite, the intersection infinite.
Those k are the elements of ℕ_def.

Regards, WM
>

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: horand.g...@gmail.com (Gus Gassmann)
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by: Gus Gassmann - Wed, 31 Aug 2022 13:03 UTC

On Wednesday, 31 August 2022 at 09:34:49 UTC-3, WM wrote:
> Gus Gassmann schrieb am Mittwoch, 31. August 2022 um 03:58:45 UTC+2:
> > On Tuesday, 30 August 2022 at 16:51:24 UTC-3, WM wrote:
> > > Gus Gassmann schrieb am Dienstag, 30. August 2022 um 19:48:23 UTC+2:
> > > > On Tuesday, 30 August 2022 at 11:59:45 UTC-3, WM wrote:
> > > > [...]
> > > > > > There is no such endsegment. So what? There is no endsegment that has an element in common with all its successors.
> > > > > Wrong. Then a successor would exist without an element in common with some predecessor.
> > > > For every end segment E(n) and every end segment E(m) there exists a natural number k in E(n) \intersect E(m).
> > > Not only one, but infinitely many.
> > One is enough.
> One alone would be too much. It was identifyable.

But it exists. Therefore no pairwise intersection of end segments, and therefore no end segment, is empty. You even stipulated that the intersection has infinite cardinality. So you are a fucking two-faced liar.

> > > > There does *NOT* exist a natural number k such that for every E(n) and E(m) [k in E(n) \intersect E(m)].
> > > What obstacle would hinder to collect as many definable endsegments as possible? Would their intersection be empty?
>
> > The fact that no end segment E(n+1) contains the minimum element n associated with E(n) makes an infinite intersection of (distinct) end segments the empty set.
> It makes the sequence of endsegments get empty too.

Nope. Quit lying! Every end segment E(n) contains at the very least two elements: n and n+1.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Wed, 31 Aug 2022 13:06 UTC

Jim Burns schrieb am Mittwoch, 31. August 2022 um 01:45:01 UTC+2:
> On 8/30/2022 3:58 PM, WM wrote:
> > All sets of two infinite endsegments
> > do not have a last endsegment.
> No.
> All sets of two (infinite) end segments
> have a last end segment.

Each one has. All together have not.

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Wed, 31 Aug 2022 13:10 UTC

zelos...@gmail.com schrieb am Mittwoch, 31. August 2022 um 07:05:13 UTC+2:
> tisdag 30 augusti 2022 kl. 15:42:23 UTC+2 skrev WM:
> > Fritz Feldhase schrieb am Montag, 29. August 2022 um 16:07:22 UTC+2:
> > > On Monday, August 29, 2022 at 3:27:52 PM UTC+2, WM wrote:
> > >
> > > > you are not aware of the fact that dark numbers exist and that your proofs are invalid for dark numbers.
> > > Yeah, I have to admit that I'm indeed not aware of these "facts".
> > But you need them to understand set theory correctly.
>
> We do not need "dark numbers" to understand them as they do not exist.

You do not understand.
>
> > If no dark numbers would blow up the set ℕ to accomplish
>
> "blow up" is not a mathematical term.

You do not know it.
>
> > ∩{E(k) : k ∈ ℕ} = { }
> > then only the elements of
> > ∀k ∈ ℕ: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀
> > would exist and therefore
> > ∩{E(k) : k ∈ ℕ} =/= { } .
> > >
> Non-sequiter galore.

You cannot follow.

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Wed, 31 Aug 2022 13:15 UTC

Gus Gassmann schrieb am Mittwoch, 31. August 2022 um 15:03:37 UTC+2:
> On Wednesday, 31 August 2022 at 09:34:49 UTC-3, WM wrote:
> > Gus Gassmann schrieb am Mittwoch, 31. August 2022 um 03:58:45 UTC+2:

> > > > > For every end segment E(n) and every end segment E(m) there exists a natural number k in E(n) \intersect E(m).
> > > > Not only one, but infinitely many.
> > > One is enough.
> > One alone would be too much. It was identifyable.
> But it exists.

Existing among dark elements does not make it existing as an individual.

> Therefore no pairwise intersection of end segments, and therefore no end segment, is empty. You even stipulated that the intersection has infinite cardinality.

The intersection of infinite endsegments!

> > > The fact that no end segment E(n+1) contains the minimum element n associated with E(n) makes an infinite intersection of (distinct) end segments the empty set.
> > It makes the sequence of endsegments get empty too.
> Nope. Quit lying! Every end segment E(n) contains at the very least two elements: n and n+1.

So the infinite loss does not concern every natnumber in the endsegments. Why should it concern every natnumber of the intersectiion?

Regards, WM

Re: Natural numbers and vases III

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From: inva...@invalid.com (Sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Wed, 31 Aug 2022 09:04:45 -0500
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by: Sergio - Wed, 31 Aug 2022 14:04 UTC

On 8/31/2022 8:02 AM, WM wrote:
> Jim Burns schrieb am Dienstag, 30. August 2022 um 22:30:29 UTC+2:
>> On 8/30/2022 3:51 PM, WM wrote:
>>
>>> What obstacle would hinder to collect
>>> as many definable endsegments as possible?
>> No obstacle. Say
>> "This collection contains all and only
>> (definable) end segments."
>
> and has an infinite intersection:
> |∩{E(1), E(2), ..., E(k)}| = ℵ₀

because you stop at k.

|∩{E(1), E(2), ...}| = 0

>>
>> Nothing can be added.
>> Nothing can be removed.
>
> That is wrong.

no, you are wrong, sets are fixed.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
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by: Sergio - Wed, 31 Aug 2022 15:01 UTC

On 8/31/2022 8:15 AM, WM wrote:
> Gus Gassmann schrieb am Mittwoch, 31. August 2022 um 15:03:37 UTC+2:
>> On Wednesday, 31 August 2022 at 09:34:49 UTC-3, WM wrote:
>>> Gus Gassmann schrieb am Mittwoch, 31. August 2022 um 03:58:45 UTC+2:
>
>>>>>> For every end segment E(n) and every end segment E(m) there exists a natural number k in E(n) \intersect E(m).
>>>>> Not only one, but infinitely many.
>>>> One is enough.
>>> One alone would be too much. It was identifyable.
>> But it exists.
>
> Existing among dark elements does not make it existing as an individual.

Red Herring.

>
>> Therefore no pairwise intersection of end segments, and therefore no end segment, is empty. You even stipulated that the intersection has infinite cardinality.
>
> The intersection of infinite endsegments!

is Empty!

>
>>>> The fact that no end segment E(n+1) contains the minimum element n associated with E(n) makes an infinite intersection of (distinct) end segments the empty set.
>>> It makes the sequence of endsegments get empty too.
>> Nope. Quit lying! Every end segment E(n) contains at the very least two elements: n and n+1.
>
> So the infinite loss does not concern every natnumber in the endsegments. Why should it concern every natnumber of the intersectiion?

that makes no sense at all.

Try using EQUATIONS.

>
> Regards, WM

tech / sci.math / Re: Natural numbers and vases III

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