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Use the Force, Luke.

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
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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
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	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	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	William
Re: Natural numbers and vases III	WM
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
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	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	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	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	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
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	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	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	WM
Re: Natural numbers and vases III	WM
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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	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	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
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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
Re: Natural numbers and vases III	William
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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	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
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	William
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	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

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Fri, 19 Aug 2022 11:56 UTC

Jim Burns schrieb am Donnerstag, 18. August 2022 um 23:27:10 UTC+2:

> The union ⋃𝓕 of FISONs is
> a collection with a counting-order
> which begins at 0

And it is finite because there cannot be more FISONs than at least one of the FISONs can distinguish.
>
> However,
> ⋃𝓕 does not end anywhere.

It does. The end is far beyond your sight. But there is no infinite FISON.

> The collection of FISON-ends is not a FISON.

Every finite FISON-end is in a FISON.
Which is the first FISON-end that is not in a FISON?

> > The number of natural numbers is
> > a natural number.
> The number of natural numbers is NOT
> a natural number,
> because
> the collection of FISON-ends is NOT
> a FISON,
> because
> 𝐹∪{n+1} is ALSO a FISON.

See? That does never end and never become larger than every FISON.

Regards, WM

Re: Natural numbers and vases III

<d2aeb31c-3003-413e-8f17-a2ba12afe4ean@googlegroups.com>

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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 - Fri, 19 Aug 2022 11:57 UTC

On Friday, 19 August 2022 at 08:42:33 UTC-3, WM wrote:
> Gus Gassmann schrieb am Donnerstag, 18. August 2022 um 16:17:58 UTC+2:
> > On Thursday, 18 August 2022 at 11:07:31 UTC-3, WM wrote:
> > [...]
> > > > The number of natural numbers is the cardinality of the set |N, namely aleph_0.
> > > The number of definable numbers is less.
> > The set of definable numbers is decidedly larger than the set of definable _natural_ numbers.
> Not even in set theory. All definable numbers need finite definitions. There are only countably many finite definitions.

Are you fucking out of your mind??? Presumably even you are not stupid enough to claim that every definable number must be a natural number!?! Maybe you have to learn to read again?

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: Fri, 19 Aug 2022 07:58:09 -0400
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by: FromTheRafters - Fri, 19 Aug 2022 11:58 UTC

WM brought next idea :
> William schrieb am Donnerstag, 18. August 2022 um 17:23:58 UTC+2:
>> On Thursday, August 18, 2022 at 11:08:13 AM UTC-3, WM wrote:
>>> William schrieb am Mittwoch, 17. August 2022 um 17:57:29 UTC+2:
>>
>>
>>>> The fact that the *set* of successors cannnot be subtracted by a stepwise
>>>> process does not mean there is an *element* of the set of successors that
>>>> cannot be subtracted by a stepwise process.
>>> There are infinitely many as is proved here Piffle. These two statements
>>> show that each finite subset of |N is followed by a set of elements with
>>> cardinality ℵo (a triviality) ∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo and
>>> here
>>>> ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo.
>>>
>>> This does never change.
>> Indeed, the fact that {1,2,3,...,n} ( or {1,2,3,...k,k+1}) is finite never
>> changes. So the two statements do not show that an infinite subset of |N is
>> followed by a set of cardintality ℵo
>
> The set of definable numbers is finite but with not upper threshold. It
> cannot be ℵo-infinite because ∀k ∈ ℕ: |ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \
> {1, 2, 3, ..., k, k+1}| = ℵo.
>
> Note: _For all natural numbers_ which are ends of FISONs.
>>
>>> But you cannot define any element that must be remaining [after "all
>>> elements" the set |N]
>>
>> There is no such element. |N is not a finite subset of |N
>
> Note above: For all derinable natural numbers. (Definable number = end of a
> FISON.)

Okay, in that case every definable natural number *without exception*
is definable/defined as being the greatest/last element of a FISON.

Seems a bit circular to me, but muckymath isn't very intuitive.

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 - Fri, 19 Aug 2022 12:02 UTC

FromTheRafters schrieb am Freitag, 19. August 2022 um 13:48:23 UTC+2:
> WM wrote :
> > FromTheRafters schrieb am Donnerstag, 18. August 2022 um 17:43:10 UTC+2:
> >> WM was thinking very hard :
> >
> >>>> The number of natural numbers is the cardinality of the set |N, namely
> >>>> aleph_0.
> >>>
> >>> The number of definable numbers is less.
> >
> >> They are already defined, all of them. As such, they were all evidently
> >> definable.
> >
> > Not individually.
> who cares?

Every accurate mathmatician.

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 - Fri, 19 Aug 2022 12:05 UTC

FromTheRafters schrieb am Freitag, 19. August 2022 um 13:47:39 UTC+2:
> WM formulated on Friday :
> > zelos...@gmail.com schrieb am Freitag, 19. August 2022 um 09:10:07 UTC+2:
> >> torsdag 18 augusti 2022 kl. 16:08:13 UTC+2 skrev WM:
> >>> William schrieb am Mittwoch, 17. August 2022 um 17:57:29 UTC+2:
> >
> >>>> The fact that the *set* of successors cannnot be subtracted by a stepwise
> >>>> process does not mean there is an *element* of the set of successors that
> >>>> cannot be subtracted by a stepwise process.
> >>> There are infinitely many as is proved here
> >>> ∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> >>> and here
> >>>> ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo.
> >>>
> >>> This does never change. But you cannot define any element that must be
> >>> remaining. That shows dark numbers.
> >>>
> >> No it doesn't! It just shows that cardinal arithmetic works different from
> >> ordinary arithmetic
> >
> > Logic works the same. This statement
> >> ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo
> > says that ℵo numbers are lurking behind all definable numbers. So they are
> > undefinable.
> How do you determine if an undefinable thing qualifies as a member of
> any particular set?

That can only implicitly be derived from the difference of actual infinity and definable numbers. Without actual infinity there are no dark nubers.

Regards, WM

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: Fri, 19 Aug 2022 08:13:22 -0400
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by: FromTheRafters - Fri, 19 Aug 2022 12:13 UTC

WM formulated on Friday :
> Jim Burns schrieb am Donnerstag, 18. August 2022 um 23:27:10 UTC+2:
>
>> The union ⋃? of FISONs is
>> a collection with a counting-order
>> which begins at 0
>
> And it is finite because there cannot be more FISONs than at least one of the
> FISONs can distinguish.
>>
>> However,
>> ⋃? does not end anywhere.
>
> It does. The end is far beyond your sight. But there is no infinite FISON.
>
>> The collection of FISON-ends is not a FISON.
>
> Every finite FISON-end is in a FISON.

Why confuse yourself with the extra 'finite'? Since you have F in FISON
which already means finite, there is no need. "Every FISON end is in a
FISON" -- well DUH! I'll even go so far as to say every FISON has two
FISON ends. The endSEGMENT of the natural order sequence is *after* the
finite initial segment and has a least element. These endsegments are
all the same size (countable, no last element) when taken as sets.
There is nothing being exhausted or distinguished whatever those terms
are supposed to mean.

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: Fri, 19 Aug 2022 08:15:55 -0400
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by: FromTheRafters - Fri, 19 Aug 2022 12:15 UTC

WM has brought this to us :
> FromTheRafters schrieb am Freitag, 19. August 2022 um 13:47:39 UTC+2:
>> WM formulated on Friday :
>>> zelos...@gmail.com schrieb am Freitag, 19. August 2022 um 09:10:07 UTC+2:
>>>> torsdag 18 augusti 2022 kl. 16:08:13 UTC+2 skrev WM:
>>>>> William schrieb am Mittwoch, 17. August 2022 um 17:57:29 UTC+2:
>>>>>> The fact that the *set* of successors cannnot be subtracted by a
>>>>>> stepwise process does not mean there is an *element* of the set of
>>>>>> successors that cannot be subtracted by a stepwise process.
>>>>> There are infinitely many as is proved here
>>>>> ∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>>>>> and here
>>>>>> ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo.
>>>>>
>>>>> This does never change. But you cannot define any element that must be
>>>>> remaining. That shows dark numbers.
>>>>>
>>>> No it doesn't! It just shows that cardinal arithmetic works different from
>>>> ordinary arithmetic
>>>
>>> Logic works the same. This statement
>>>> ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo
>>> says that ℵo numbers are lurking behind all definable numbers. So they are
>>> undefinable.
>> How do you determine if an undefinable thing qualifies as a member of
>> any particular set?
>
> That can only implicitly be derived from the difference of actual infinity
> and definable numbers. Without actual infinity there are no dark nubers.

IOW, no clue huh?

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: Fri, 19 Aug 2022 08:17:27 -0400
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by: FromTheRafters - Fri, 19 Aug 2022 12:17 UTC

WM presented the following explanation :
> FromTheRafters schrieb am Freitag, 19. August 2022 um 13:48:23 UTC+2:
>> WM wrote :
>>> FromTheRafters schrieb am Donnerstag, 18. August 2022 um 17:43:10 UTC+2:
>>>> WM was thinking very hard :
>>>
>>>>>> The number of natural numbers is the cardinality of the set |N, namely
>>>>>> aleph_0.
>>>>>
>>>>> The number of definable numbers is less.
>>>> They are already defined, all of them. As such, they were all evidently
>>>> definable.
>>>
>>> Not individually.
>> who cares?
>
> Every accurate mathmatician.

Show me one and I'll wager he or she is also ignorant of induction.

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Subject: Re: Natural numbers and vases III
From: wpihug...@gmail.com (William)
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by: William - Fri, 19 Aug 2022 12:31 UTC

On Friday, August 19, 2022 at 8:48:16 AM UTC-3, WM wrote:

> The set of numbers [that can be written down] is finite but with not upper threshold.

Absolute nonsense. There is no such thing as a finite set with no upper threshold.

It cannot be ℵo-infinite because
> ∀k ∈ ℕ: |ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo.

So what? A statement that every *element* of |N has some property, does not imply that the *set* has this property.

--
William Hughes

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: Fri, 19 Aug 2022 07:45:56 -0500
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by: Sergio - Fri, 19 Aug 2022 12:45 UTC

On 8/19/2022 6:48 AM, WM wrote:
> William schrieb am Donnerstag, 18. August 2022 um 17:23:58 UTC+2:
>> On Thursday, August 18, 2022 at 11:08:13 AM UTC-3, WM wrote:
>>> William schrieb am Mittwoch, 17. August 2022 um 17:57:29 UTC+2:
>>
>>
>>>> The fact that the *set* of successors cannnot be subtracted by a stepwise process does not mean there is an *element* of the set of successors that cannot be subtracted by a stepwise process.
>>> There are infinitely many as is proved here
>> Piffle. These two statements show that each finite subset of |N is followed by a set of elements with cardinality ℵo (a triviality)
>>> ∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>>> and here
>>> |ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo.
>>>
>>> This does never change.
>> Indeed, the fact that {1,2,3,...,n} ( or {1,2,3,...k,k+1}) is finite never changes. So the two statements do not show that an infinite subset of |N is followed by a set of cardintality ℵo
>
> The set of definable numbers is finite but with not upper threshold.

Wrong. What is wrong with you ?

> It cannot be ℵo-infinite because
> ∀k ∈ ℕ: |ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo.

the equn is right, however your use of it is wrong. The upper threshold is k, and then k+1.

>
> Note: _For all natural numbers_ which are ends of FISONs.

Wrong, FISONs are composed of natural numbers.

>>
>>> But you cannot define any element that must be remaining [after "all elements" the set |N]
>>
>> There is no such element. |N is not a finite subset of |N
>
> Note above: For all derinable natural numbers. (Definable number = end of a FISON.)

Wrong, your daffynition is a "definable" number is using raps, taps, beeps, flashes of light, colors of the rainbow...

note that by your daffynition, you equate FISONs to raps, taps, colors of the rainbow,

>
> 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: Fri, 19 Aug 2022 07:50:07 -0500
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by: Sergio - Fri, 19 Aug 2022 12:50 UTC

On 8/19/2022 6:39 AM, WM wrote:
> FromTheRafters schrieb am Donnerstag, 18. August 2022 um 17:43:10 UTC+2:
>> WM was thinking very hard :
>
>>>> The number of natural numbers is the cardinality of the set |N, namely
>>>> aleph_0.
>>>
>>> The number of definable numbers is less.
>
>> They are already defined, all of them. As such, they were all evidently
>> definable.
>
> Not individually. Behind all individually definable numbers there are almost all numbers. They are not individually definable as
> ∀k ∈ ℕ_deef: |ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo

all that equation says is that endsegments are infinite, GOT THAT ?

> proves.
>
> Why do you refuse logic?

it is lies and crap, not logic.

your "logic" is defective.

>
> Regards, WM

Re: Natural numbers and vases III

<5ac2bc5c-d5c8-48cf-818c-96e7bd256f48@att.net>

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Fri, 19 Aug 2022 12:52:17 -0400
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by: Jim Burns - Fri, 19 Aug 2022 16:52 UTC

On 8/19/2022 7:56 AM, WM wrote:
> Jim Burns schrieb am Donnerstag,
> 18. August 2022 um 23:27:10 UTC+2:

>> The union ⋃𝓕 of FISONs is
>> a collection with a counting-order
>> which begins at 0
>
> And it is finite because
> there cannot be more FISONs than
> at least one of the FISONs can distinguish.

>>> The number of natural numbers is
>>> a natural number.
>>
>> The number of natural numbers is NOT
>> a natural number,
>> because
>> the collection of FISON-ends is NOT
>> a FISON,
>> because
>> 𝐹∪{n+1} is ALSO a FISON.
>
> See?
> That does never end and
> never become larger than every FISON.

> there cannot be more FISONs than
> at least one of the FISONs can distinguish.

No.
There are more FISONs than
any FISON can distinguish.

If
𝐹 is a FISON,
then
𝐹 is
a collection with a counting-order
which begins at 0
and which ends _somewhere_

If
𝐹 is a FISON, and 𝐹 ends at n,
then
𝐹∪{n+1} is also
a collection with a counting-order
which begins at 0
and which ends somewhere: at n+1,
and
𝐹∪{n+1} is also a FISON,
and
𝐹 cannot distinguish
all the FISONs with ends in 𝐹
and 𝐹∪{n+1} as well.

If
𝐹 is a FISON,
then
𝐹 cannot distinguish all the FISONs.

----
For each n in any FISON,
n+1 is also in a FISON,
sometimes a different FISON.

∀𝐹 ∈ 𝓕, ∀n ∈ 𝐹, ∃𝐹' ∈ 𝓕: n+1 ∈ 𝐹'
| because
| 𝐹ₙ∪{n+1} is ALSO a FISON.

There is NO n in the union ⋃𝓕 of FISONs
for which n+1 is NOT also in ⋃𝓕
∀n ∈ ⋃𝓕: n+1 ∈ ⋃𝓕
~(∃n ∈ ⋃𝓕: n+1 ∉ ⋃𝓕)

Each FISON ends somewhere, unlike ⋃𝓕
∀𝐹 ∈ 𝓕: (∃n ∈ 𝐹: n+1 ∉ F)

Each FISON is not ⋃𝓕
⋃𝓕 is not a FISON
∀𝐹 ∈ 𝓕: 𝐹 ≠ ⋃𝓕
⋃𝓕 ∉ 𝓕

𝓔 = { ⋃𝓕\𝐹 : 𝐹 ∈ 𝓕 }
∀𝐹 ∈ 𝓕: ⋃𝓕\𝐹 ≠ ∅
∅ ∉ 𝓔

⋂{𝑌\𝑋 : 𝑋 ∈ 𝓧} = 𝑌\⋃𝓧
⋂𝓔 = ⋃𝓕\⋃𝓕 = ∅

⋂𝓔 = ∅ ∧ ∅ ∉ 𝓔
| because
| 𝐹ₙ∪{n+1} is ALSO a FISON.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
Injection-Date: Sat, 20 Aug 2022 13:48:50 +0000
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by: WM - Sat, 20 Aug 2022 13:48 UTC

William schrieb am Freitag, 19. August 2022 um 14:31:56 UTC+2:
> On Friday, August 19, 2022 at 8:48:16 AM UTC-3, WM wrote:
>
>
> > The set of numbers [that can be written down] is finite but with not upper threshold.
>
> Absolute nonsense. There is no such thing as a finite set with no upper threshold.

What is the upper threshold of the set of n defined by
|ℕ \ {1, 2, 3, ..., n}| = ℵo ?
It cannot be ℵo-infinite because ℵo elements are following.

> It cannot be ℵo-infinite because
> > ∀k ∈ ℕ: |ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo.
> So what? A statement that every *element* of |N has some property, does not imply that the *set* has this property.

I am not interested in the set but only in all its elements.

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 - Sat, 20 Aug 2022 13:51 UTC

Jim Burns schrieb am Freitag, 19. August 2022 um 18:52:27 UTC+2:
> On 8/19/2022 7:56 AM, WM wrote:
> > That does never end and
> > never become larger than every FISON.
> > there cannot be more FISONs than
> > at least one of the FISONs can distinguish.
> No.
> There are more FISONs than
> any FISON can distinguish.

Then there is a surplus that cannot distinguished.

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 - Sat, 20 Aug 2022 14:05 UTC

On Saturday, 20 August 2022 at 10:48:54 UTC-3, WM wrote:
[...]
> What is the upper threshold of the set of n defined by
> |ℕ \ {1, 2, 3, ..., n}| = ℵo ? (*)
> It cannot be ℵo-infinite because ℵo elements are following.

That's a blatant non sequitur. Precisely /because/ there are aleph_0 natural numbers larger than any n (specifically: n+1), there is no largest n satisfying (*). You have tried induction and failed, you have tried cardinality and failed, you have tried switching quantifiers and failed, you have tried inclusion monotony and failed. So you are either utterly ignorant, or you are lying cheat, or both.

But of course there is an upper threshold: omega. It is the smallest upper bound, trivially.

Re: Natural numbers and vases III

<tdqpuu$1u1t$1@gioia.aioe.org>

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From: inva...@invalid.com (Sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Sat, 20 Aug 2022 09:09:31 -0500
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by: Sergio - Sat, 20 Aug 2022 14:09 UTC

On 8/20/2022 8:48 AM, WM wrote:
> William schrieb am Freitag, 19. August 2022 um 14:31:56 UTC+2:
>> On Friday, August 19, 2022 at 8:48:16 AM UTC-3, WM wrote:
>>
>>
>>> The set of numbers [that can be written down] is finite but with not upper threshold.
>>
>> Absolute nonsense. There is no such thing as a finite set with no upper threshold.
>
> What is the upper threshold of the set of n defined by
> |ℕ \ {1, 2, 3, ..., n}| = ℵo ?

that is not a set. It is a count. It is a count of the set Endsegment(n+1).

> It cannot be ℵo-infinite because ℵo elements are following.

That statement is conformation you cannot read or use equations, troll.

>
>> It cannot be ℵo-infinite because
>>> ∀k ∈ ℕ: |ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo.
>> So what? A statement that every *element* of |N has some property, does not imply that the *set* has this property.
>
> I am not interested in the set but only in all its elements.

That is because you do not understand what sets mean, Pretender.

>
> Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
Date: Sat, 20 Aug 2022 09:10:44 -0500
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by: Sergio - Sat, 20 Aug 2022 14:10 UTC

On 8/20/2022 8:51 AM, WM wrote:
> Jim Burns schrieb am Freitag, 19. August 2022 um 18:52:27 UTC+2:
>> On 8/19/2022 7:56 AM, WM wrote:
>
>>> That does never end and
>>> never become larger than every FISON.
>>> there cannot be more FISONs than
>>> at least one of the FISONs can distinguish.
>> No.
>> There are more FISONs than
>> any FISON can distinguish.
>
> Then there is a surplus that cannot distinguished.
>
> Regards, WM

NEW ANTS !!!

Surplus Ants
Surplus Ants that cannot distinguished.

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 - Sat, 20 Aug 2022 14:30 UTC

On Saturday, August 20, 2022 at 10:48:54 AM UTC-3, WM wrote:

> I am not interested in the set but only in all its elements.

Piffle "all its elements" is the set.

--
William Hughes

Re: Natural numbers and vases III

<tdqrn4$1upms$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: Sat, 20 Aug 2022 10:39:25 -0400
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by: FromTheRafters - Sat, 20 Aug 2022 14:39 UTC

WM expressed precisely :
> William schrieb am Freitag, 19. August 2022 um 14:31:56 UTC+2:
>> On Friday, August 19, 2022 at 8:48:16 AM UTC-3, WM wrote:
>>
>>
>>> The set of numbers [that can be written down] is finite but with not upper
>>> threshold.
>>
>> Absolute nonsense. There is no such thing as a finite set with no upper
>> threshold.

Threshold? Are we making hay now?

> What is the upper threshold of the set of n defined by
>> ℕ \ {1, 2, 3, ..., n}| = ℵo ?
> It cannot be ℵo-infinite because ℵo elements are following.

Balderdash! It is exactly why. There is no last element and you are
taking only a finite subset of it out of consideration in your new
'still inductive' set.

>> It cannot be ℵo-infinite because
>>> ∀k ∈ ℕ: |ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo.
>> So what? A statement that every *element* of |N has some property, does not
>> imply that the *set* has this property.
>
> I am not interested in the set but only in all its elements.

How do you determine if a particular element qualifies to be in the
set?

Re: Natural numbers and vases III

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From: FTR...@nomail.afraid.org (FromTheRafters)
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Subject: Re: Natural numbers and vases III
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by: FromTheRafters - Sat, 20 Aug 2022 14:56 UTC

William was thinking very hard :
> On Saturday, August 20, 2022 at 10:48:54 AM UTC-3, WM wrote:
>
>> I am not interested in the set but only in all its elements.
>
> Piffle "all its elements" is the set.

Not all muckysets are sets in the ZFC sense. He has a muckyset of
counting numbers some of which have been 'used' and are thus 'defined'
and some which have not been used yet which are 'undefined' and some
that are so huge that he calls them 'undefinable' as if they could
never conceivably be used.

"All muckyset elements" is not necessarily 'the (ZFC) set' -- that is
to say that 'all elements of a collection of objects some of which are
undefinable' is not a set in ZFC despite that fact that everything *in*
ZFC is a set.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Sat, 20 Aug 2022 15:03 UTC

On Saturday, August 20, 2022 at 3:48:54 PM UTC+2, WM was talking about

> the set of n [e IN] defined by |ℕ \ {1, 2, 3, ..., n}| = ℵo

Well, using the language of set theory, you are considering the set

{n e IN : |ℕ \ {1, 2, 3, ..., n}| = ℵo}

it seems. Since

An e IN: |ℕ \ {1, 2, 3, ..., n}| = ℵo

is a theorem, we get that

{n e IN : |ℕ \ {1, 2, 3, ..., n}| = ℵo} = IN.

> It cannot be ℵo-infinite because ℵo elements are following.

Yeah: "[WM’s] conclusions are based on the sloppiness of his notions, his inability of giving precise definitions, his fundamental misunderstanding of elementary mathematical concepts, and sometimes, as the late Dik Winter remarked [...], on nothing at all."

Actually, "it" is infinit and has cardinality ℵo, since IN is infinite and has cardinality ℵo.

> I am not interested in the set but only in all its elements.

Huh?! Please make up your mind, you psychotic asshole:

FIRST you talked about "the set of n e IN ... defined by |ℕ \ {1, 2, 3, ..., n}| = ℵo",
NOW you claim that you are "not interested in the set but only in all its elements".

Holy shit!

Hint: There are countably-infinitely many natural numbers in IN.

Re: Natural numbers and vases III

<tdqt6a$1v6po$1@dont-email.me>

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Subject: Re: Natural numbers and vases III
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by: FromTheRafters - Sat, 20 Aug 2022 15:04 UTC

Sergio formulated the question :
> On 8/20/2022 8:51 AM, WM wrote:
>> Jim Burns schrieb am Freitag, 19. August 2022 um 18:52:27 UTC+2:
>>> On 8/19/2022 7:56 AM, WM wrote:
>>
>>>> That does never end and
>>>> never become larger than every FISON.
>>>> there cannot be more FISONs than
>>>> at least one of the FISONs can distinguish.
>>> No.
>>> There are more FISONs than
>>> any FISON can distinguish.
>>
>> Then there is a surplus that cannot distinguished.
>>
>> Regards, WM
>
> NEW ANTS !!!
>
> Surplus Ants
> Surplus Ants that cannot distinguished.

Threshold Ants

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Sat, 20 Aug 2022 15:07 UTC

On Saturday, August 20, 2022 at 4:30:41 PM UTC+2, William wrote:
> On Saturday, August 20, 2022 at 10:48:54 AM UTC-3, WM wrote:
> >
> > I am not interested in the set but only in all its elements.
> >
> Piffle "all its elements" is the set.

No, it is't, you silly asshole.

Hint: Consider the set {1, 2, 3}. The numbers 1, 2, and 3 are "all its elements". But the numbers 1, 2, and 3 AREN'T the set {1, 2, 3}.

Actually, all elements in {1, 2, 3} are natural numbers, but the set {1, 2, 3} is NOT a natural number.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Sat, 20 Aug 2022 15:07 UTC

On Saturday, August 20, 2022 at 5:04:50 PM UTC+2, FromTheRafters wrote:
> Sergio formulated the question :
> > On 8/20/2022 8:51 AM, WM wrote:
> >> Jim Burns schrieb am Freitag, 19. August 2022 um 18:52:27 UTC+2:
> >>> On 8/19/2022 7:56 AM, WM wrote:
> >>
> >>>> That does never end and
> >>>> never become larger than every FISON.
> >>>> there cannot be more FISONs than
> >>>> at least one of the FISONs can distinguish.
> >>> No.
> >>> There are more FISONs than
> >>> any FISON can distinguish.
> >>
> >> Then there is a surplus that cannot distinguished.
> >>
> >> Regards, WM
> >
> > NEW ANTS !!!
> >
> > Surplus Ants
> > Surplus Ants that cannot distinguished.
> >
> Threshold Ants

All its elements Ants

Re: Natural numbers and vases III

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From: FTR...@nomail.afraid.org (FromTheRafters)
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Subject: Re: Natural numbers and vases III
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by: FromTheRafters - Sat, 20 Aug 2022 15:39 UTC

Fritz Feldhase expressed precisely :
> On Saturday, August 20, 2022 at 4:30:41 PM UTC+2, William wrote:
>> On Saturday, August 20, 2022 at 10:48:54 AM UTC-3, WM wrote:
>>>
>>> I am not interested in the set but only in all its elements.
>>>
>> Piffle "all its elements" is the set.
>
> No, it is't, you silly asshole.
>
> Hint: Consider the set {1, 2, 3}. The numbers 1, 2, and 3 are "all its
> elements". But the numbers 1, 2, and 3 AREN'T the set {1, 2, 3}.
>
> Actually, all elements in {1, 2, 3} are natural numbers, but the set {1, 2,
> 3} is NOT a natural number.

I believe it is the natural number 3, under Frege's construction
matching (equinumerosity) cardinal numbers to the naturals. Of course,
WM and William were not thinking about this, I assume.

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

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