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

SubjectAuthor
* Natural numbers and vasesWilliam
+- Re: Natural numbers and vasesRoss A. Finlayson
+* Re: Natural numbers and vasesWM
|+* Re: Natural numbers and vasesWilliam
||`* Re: Natural numbers and vasesWM
|| +- Re: Natural numbers and vasessergio
|| `* Re: Natural numbers and vasesWilliam
||  `* Re: Natural numbers and vasesWM
||   +- Re: Natural numbers and vasessergio
||   `* Re: Natural numbers and vasesWilliam
||    `* Re: Natural numbers and vasesWM
||     +- Re: Natural numbers and vasessergio
||     `* Re: Natural numbers and vasesWilliam
||      +- Re: Natural numbers and vasesFritz Feldhase
||      `* Re: Natural numbers and vasesWM
||       +* Re: Natural numbers and vasesWilliam
||       |+* Re: Natural numbers and vasesGus Gassmann
||       ||`* Re: Natural numbers and vasesFritz Feldhase
||       || `- Re: Natural numbers and vasesGus Gassmann
||       |`* Re: Natural numbers and vasesWM
||       | +- Re: Natural numbers and vasesGus Gassmann
||       | +* Re: Natural numbers and vasesWilliam
||       | |+* Re: Natural numbers and vasesWM
||       | ||+* Re: Natural numbers and vasesWilliam
||       | |||`* Re: Natural numbers and vasesWM
||       | ||| `* Re: Natural numbers and vasesWilliam
||       | |||  +* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |`- Re: Natural numbers and vasessergio
||       | |||  +* Re: Natural numbers and vasesGus Gassmann
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||+- Re: Natural numbers and vasesFromTheRafters
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesGus Gassmann
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesGus Gassmann
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesGus Gassmann
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesGus Gassmann
||       | |||  ||`- Re: Natural numbers and vasesChris M. Thomasson
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesGus Gassmann
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||+* Re: Natural numbers and vasesJim Burns
||       | |||  |||`* Re: Natural numbers and vasesWM
||       | |||  ||| +* Re: Natural numbers and vasesJim Burns
||       | |||  ||| |`* Re: Natural numbers and vasesWM
||       | |||  ||| | +* Re: Natural numbers and vasesJim Burns
||       | |||  ||| | |`* Re: Natural numbers and vasesWM
||       | |||  ||| | | +* Re: Natural numbers and vasesJim Burns
||       | |||  ||| | | |`* Re: Natural numbers and vasesWM
||       | |||  ||| | | | `- Re: Natural numbers and vasessergio
||       | |||  ||| | | `- Re: Natural numbers and vasessergio
||       | |||  ||| | `- Re: Natural numbers and vasessergio
||       | |||  ||| `- Re: Natural numbers and vasessergio
||       | |||  ||`* Re: Natural numbers and vasessergio
||       | |||  || `- Re: Natural numbers and vasesChet Hirasi
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+* Re: Natural numbers and vasesGus Gassmann
||       | |||  ||`- Re: Natural numbers and vasesChet Hirasi
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`* Re: Natural numbers and vasessergio
||       | |||  || `* Re: Natural numbers and vasesWM
||       | |||  ||  `* Re: Natural numbers and vasessergio
||       | |||  ||   +* Re: Natural numbers and vasessergio
||       | |||  ||   |`* Re: Natural numbers and vasesFredJeffries
||       | |||  ||   | `- Re: Natural numbers and vasessergio
||       | |||  ||   `* Re: Natural numbers and vasesChris M. Thomasson
||       | |||  ||    `* Re: Natural numbers and vasessergio
||       | |||  ||     +* Re: Natural numbers and vasesChris M. Thomasson
||       | |||  ||     |`- Re: Natural numbers and vasessergio
||       | |||  ||     `* Re: Natural numbers and vasesChris M. Thomasson
||       | |||  ||      `* Re: Natural numbers and vasessergio
||       | |||  ||       `* Re: Natural numbers and vasesChris M. Thomasson
||       | |||  ||        +* Re: Natural numbers and vasesChris M. Thomasson
||       | |||  ||        |`- Re: Natural numbers and vasessergio
||       | |||  ||        `- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||`- Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  ||+- Re: Natural numbers and vasessergio
||       | |||  ||`* Re: Natural numbers and vasesChris M. Thomasson
||       | |||  || +* Re: Natural numbers and vasesFromTheRafters
||       | |||  || `* Re: Natural numbers and vasessergio
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesGus Gassmann
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesGus Gassmann
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesGus Gassmann
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWilliam
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |+- Re: Natural numbers and vasesWM
||       | |||  |+* Re: Natural numbers and vasesWM
||       | |||  |`* Re: Natural numbers and vasesWM
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vaseszelos...@gmail.com
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vaseszelos...@gmail.com
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vaseszelos...@gmail.com
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vaseszelos...@gmail.com
||       | |||  +- Re: Natural numbers and vaseszelos...@gmail.com
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vasesFritz Feldhase
||       | |||  +- Re: Natural numbers and vaseszelos...@gmail.com
||       | |||  `- Re: Natural numbers and vaseszelos...@gmail.com
||       | ||`- Re: Natural numbers and vasessergio
||       | |`* Re: Natural numbers and vasesWM
||       | +- Re: Natural numbers and vasessergio
||       | `* Re: Natural numbers and vasesJim Burns
||       `- Re: Natural numbers and vasessergio
|+- Re: Natural numbers and vasesRoss A. Finlayson
|`- Re: Natural numbers and vasesFritz Feldhase
+* Re: Natural numbers and vasesChris M. Thomasson
+* Re: Natural numbers and vasesMostowski Collapse
`- Re: Natural numbers and vasesSocratis T.n.p.

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Re: Natural numbers and vases

<t6bjb4$1mg3$1@gioia.aioe.org>

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https://www.novabbs.com/tech/article-flat.php?id=100658&group=sci.math#100658

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From: inva...@invalid.com (sergi o)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Sat, 21 May 2022 15:50:44 -0500
Organization: Aioe.org NNTP Server
Message-ID: <t6bjb4$1mg3$1@gioia.aioe.org>
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 by: sergi o - Sat, 21 May 2022 20:50 UTC

On 5/21/2022 3:37 PM, WM wrote:
> William schrieb am Samstag, 21. Mai 2022 um 18:08:14 UTC+2:
>> On Saturday, May 21, 2022 at 1:00:38 PM UTC-3, WM wrote:
>>> William schrieb am Samstag, 21. Mai 2022 um 09:22:27 UTC+2:
>>>> On Friday, May 20, 2022 at 8:07:53 AM UTC-3, WM wrote:
>>>>> ... not any has less means all together have not less.
>>>>
>>>> Nope. Trivial counterexample Consider a Peano set S. Every element of S has a successor in S. S does not have a successor in S.
>>> There is no set S
>> Piffle. S exists by the Axiom of infinity, It is well known that you do not like the results of taking this axiom, but results that you do no like are not contradictions.
>
> The square circle exists by a similarly invalid axiom. Here are some more:
>  Axiom of two even primes: There are two even prime numbers.
>  Axiom of prime number triples: There is a second triple of prime numbers, besides (3, 5, 7).
>  Axiom of meagre sum: There is a set of n different positive natural numbers with sum n2/2.
> How obvious a contradiction has to result from an additional axiom in order to reject it?
>
> Regards, WM

how are those axioms ?

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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 by: William - Sat, 21 May 2022 21:47 UTC

On Saturday, May 21, 2022 at 5:37:47 PM UTC-3, WM wrote:

> How obvious a contradiction has to result from an additional axiom in order to reject it?

It does not have to be obvious. It must be a contradiction. The Axiom of Infinity does not lead to a contradiction it leads to results you do not like,

--
William Hughes

Re: Natural numbers and vases

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From: inva...@invalid.com (sergi o)
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Subject: Re: Natural numbers and vases
Date: Sat, 21 May 2022 17:38:24 -0500
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 by: sergi o - Sat, 21 May 2022 22:38 UTC

On 5/21/2022 3:45 PM, sergi o wrote:
> On 5/21/2022 3:32 PM, WM wrote:
>> FromTheRafters schrieb am Samstag, 21. Mai 2022 um 20:35:55 UTC+2:
>>> WM explained on 5/21/2022 :
>>>> FromTheRafters schrieb am Freitag, 20. Mai 2022 um 13:23:46 UTC+2:
>>>>> WM was thinking very hard :
>>>>
>>>>>> You are outside of logic. The set is nothing else but all its elements.
>>>>> You have said this before, do you have a citation for that?
>>>>
>>>> Unter einer "Menge" verstehen wir jede Zusammenfassung M von bestimmten
>>>> wohlunterschiedenen Objekten m unsrer Anschauung oder unseres Denkens (welche
>>>> die "Elemente" von M genannt werden) zu einem Ganzen.
>>>>
>>>> "every collection of definite well-distinguished objects of our visualization
>>>> or thinking". [Cantor]
>>
>>> "By a set we understand every combination M of certain
>>> well-distinguished objects m our intuition or our thinking (which are
>>> called the elements of M ) into a whole". This is how Georg Cantor 's
>>> contributions to the establishment of set theory , published in 1895,
>>> begin . This “naive” definition should suffice as a starting point.
>>>
>>> [...]
>>> ===================================================================
>>>
>>> Nothing about 'nothing else'.
>>
>> Of course not, since there is nothing else.
>>
>>> A bunch of 'would be' elements are not a set.
>>
>> Which bunch of finitely many elements would not be a set?
>
> why would it ?
>
> your elements are all over the floor!
> (yes I realize that could make up a set of elements on the floor, but I haven't declared it as being a set yet (HINT))
>
> Extra Credit :  how many different sets would a "bunch" of elements make ?
>
>
>        [(HINT:"bunch"!)]
>
>
> If I assemble a set of elements, then it would be a set.
>
> finite or infinite is irrelevant.
>
>
>>
>> Regards, WM
>

Extra extra credit: [(HINT:"bunch"! + 1)] why?

Re: Natural numbers and vases

<t6bq14$87j$1@dont-email.me>

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From: chris.m....@gmail.com (Chris M. Thomasson)
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Subject: Re: Natural numbers and vases
Date: Sat, 21 May 2022 15:44:50 -0700
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 by: Chris M. Thomasson - Sat, 21 May 2022 22:44 UTC

On 5/21/2022 1:37 PM, WM wrote:
> William schrieb am Samstag, 21. Mai 2022 um 18:08:14 UTC+2:
>> On Saturday, May 21, 2022 at 1:00:38 PM UTC-3, WM wrote:
>>> William schrieb am Samstag, 21. Mai 2022 um 09:22:27 UTC+2:
>>>> On Friday, May 20, 2022 at 8:07:53 AM UTC-3, WM wrote:
>>>>> ... not any has less means all together have not less.
>>>>
>>>> Nope. Trivial counterexample Consider a Peano set S. Every element of S has a successor in S. S does not have a successor in S.
>>> There is no set S
>> Piffle. S exists by the Axiom of infinity, It is well known that you do not like the results of taking this axiom, but results that you do no like are not contradictions.
>
> The square circle exists by a similarly invalid axiom. Here are some more:
>  Axiom of two even primes: There are two even prime numbers.
>  Axiom of prime number triples: There is a second triple of prime numbers, besides (3, 5, 7).
>  Axiom of meagre sum: There is a set of n different positive natural numbers with sum n2/2.
> How obvious a contradiction has to result from an additional axiom in order to reject it?

The square circle? Fwiw, here is a trigonometric version of a parametric
square:

x(t) = cos(t) * abs(cos(t))
y(t) = sin(t) * abs(sin(t))

t from 0 to pi2

;^)

In wolfram:

https://www.wolframalpha.com/input?i=x+%3D+cos%28t%29+*+abs%28cos%28t%29%29%2C+y+%3D+sin%28t%29+*+abs%28sin%28t%29%29%2C+t+from+0+to+pi2

Re: Natural numbers and vases

<t6c3gd$ohc$1@dont-email.me>

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From: nom...@afraid.org (FromTheRafters)
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Subject: Re: Natural numbers and vases
Date: Sat, 21 May 2022 18:26:30 -0700
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 by: FromTheRafters - Sun, 22 May 2022 01:26 UTC

WM presented the following explanation :
> FromTheRafters schrieb am Samstag, 21. Mai 2022 um 20:35:55 UTC+2:
>> WM explained on 5/21/2022 :
>>> FromTheRafters schrieb am Freitag, 20. Mai 2022 um 13:23:46 UTC+2:
>>>> WM was thinking very hard :
>>>
>>>>> You are outside of logic. The set is nothing else but all its elements.
>>>> You have said this before, do you have a citation for that?
>>>
>>> Unter einer "Menge" verstehen wir jede Zusammenfassung M von bestimmten
>>> wohlunterschiedenen Objekten m unsrer Anschauung oder unseres Denkens
>>> (welche die "Elemente" von M genannt werden) zu einem Ganzen.
>>>
>>> "every collection of definite well-distinguished objects of our
>>> visualization or thinking". [Cantor]
>
>> "By a set we understand every combination M of certain
>> well-distinguished objects m our intuition or our thinking (which are
>> called the elements of M ) into a whole". This is how Georg Cantor 's
>> contributions to the establishment of set theory , published in 1895,
>> begin . This “naive” definition should suffice as a starting point.
>>
>> [...]
>> ===================================================================
>>
>> Nothing about 'nothing else'.
>
> Of course not, since there is nothing else.
>
>> A bunch of 'would be' elements are not a set.
>
> Which bunch of finitely many elements would not be a set?

Those which are not set-like. Proper Classes for instance.

Re: Natural numbers and vases

<t6c417$r3e$1@dont-email.me>

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From: nom...@afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Sat, 21 May 2022 18:35:28 -0700
Organization: Peripheral Visions
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 by: FromTheRafters - Sun, 22 May 2022 01:35 UTC

sergi o used his keyboard to write :
> On 5/21/2022 3:45 PM, sergi o wrote:
>> On 5/21/2022 3:32 PM, WM wrote:
>>> FromTheRafters schrieb am Samstag, 21. Mai 2022 um 20:35:55 UTC+2:
>>>> WM explained on 5/21/2022 :
>>>>> FromTheRafters schrieb am Freitag, 20. Mai 2022 um 13:23:46 UTC+2:
>>>>>> WM was thinking very hard :
>>>>>
>>>>>>> You are outside of logic. The set is nothing else but all its
>>>>>>> elements.
>>>>>> You have said this before, do you have a citation for that?
>>>>>
>>>>> Unter einer "Menge" verstehen wir jede Zusammenfassung M von bestimmten
>>>>> wohlunterschiedenen Objekten m unsrer Anschauung oder unseres Denkens
>>>>> (welche
>>>>> die "Elemente" von M genannt werden) zu einem Ganzen.
>>>>>
>>>>> "every collection of definite well-distinguished objects of our
>>>>> visualization
>>>>> or thinking". [Cantor]
>>>
>>>> "By a set we understand every combination M of certain
>>>> well-distinguished objects m our intuition or our thinking (which are
>>>> called the elements of M ) into a whole". This is how Georg Cantor 's
>>>> contributions to the establishment of set theory , published in 1895,
>>>> begin . This “naive” definition should suffice as a starting point.
>>>>
>>>> [...]
>>>> ===================================================================
>>>>
>>>> Nothing about 'nothing else'.
>>>
>>> Of course not, since there is nothing else.
>>>
>>>> A bunch of 'would be' elements are not a set.
>>>
>>> Which bunch of finitely many elements would not be a set?
>>
>> why would it ?
>>
>> your elements are all over the floor!
>> (yes I realize that could make up a set of elements on the floor, but I
>> haven't declared it as being a set yet (HINT))
>>
>> Extra Credit :  how many different sets would a "bunch" of elements make ?
>>
>>
>>        [(HINT:"bunch"!)]
>>
>>
>> If I assemble a set of elements, then it would be a set.
>>
>> finite or infinite is irrelevant.
>>
>>
>>>
>>> Regards, WM
>>
>
> Extra extra credit: [(HINT:"bunch"! + 1)] why?

Because there's more to a set than just its elements?

Maybe he can't see it because it is dark.

Re: Natural numbers and vases

<t6c45p$rk9$1@dont-email.me>

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From: nom...@afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Sat, 21 May 2022 18:37:54 -0700
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 by: FromTheRafters - Sun, 22 May 2022 01:37 UTC

WM formulated on Saturday :
> William schrieb am Samstag, 21. Mai 2022 um 18:08:14 UTC+2:
>> On Saturday, May 21, 2022 at 1:00:38 PM UTC-3, WM wrote:
>>> William schrieb am Samstag, 21. Mai 2022 um 09:22:27 UTC+2:
>>>> On Friday, May 20, 2022 at 8:07:53 AM UTC-3, WM wrote:
>>>>> ... not any has less means all together have not less.
>>>>
>>>> Nope. Trivial counterexample Consider a Peano set S. Every element of S
>>>> has a successor in S. S does not have a successor in S.
>>> There is no set S
>> Piffle. S exists by the Axiom of infinity, It is well known that you do not
>> like the results of taking this axiom, but results that you do no like are
>> not contradictions.
>
> The square circle exists by a similarly invalid axiom. Here are some more:
>  Axiom of two even primes: There are two even prime numbers.
>  Axiom of prime number triples: There is a second triple of prime numbers,
> besides (3, 5, 7).  Axiom of meagre sum: There is a set of n different
> positive natural numbers with sum n2/2. How obvious a contradiction has to
> result from an additional axiom in order to reject it?

What is an axiom again?

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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 by: WM - Sun, 22 May 2022 13:20 UTC

FromTheRafters schrieb am Sonntag, 22. Mai 2022 um 03:38:11 UTC+2:
> WM formulated on Saturday :
> > William schrieb am Samstag, 21. Mai 2022 um 18:08:14 UTC+2:
> >> On Saturday, May 21, 2022 at 1:00:38 PM UTC-3, WM wrote:
> >>> William schrieb am Samstag, 21. Mai 2022 um 09:22:27 UTC+2:
> >>>> On Friday, May 20, 2022 at 8:07:53 AM UTC-3, WM wrote:
> >>>>> ... not any has less means all together have not less.
> >>>>
> >>>> Nope. Trivial counterexample Consider a Peano set S. Every element of S
> >>>> has a successor in S. S does not have a successor in S.
> >>> There is no set S
> >> Piffle. S exists by the Axiom of infinity, It is well known that you do not
> >> like the results of taking this axiom, but results that you do no like are
> >> not contradictions.
> >
> > The square circle exists by a similarly invalid axiom. Here are some more:
> >  Axiom of two even primes: There are two even prime numbers.
> >  Axiom of prime number triples: There is a second triple of prime numbers,
> > besides (3, 5, 7).  Axiom of meagre sum: There is a set of n different
> > positive natural numbers with sum n2/2. How obvious a contradiction has to
> > result from an additional axiom in order to reject it?
> What is an axiom again?

Depends. A good axiom is a brief statement describing well-established fact.. Example: Group axioms. A bad axiom is arbitrary nonsense, often contradicting well-established fact. Example: Large cardinal axioms.

Regards, WM

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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 by: WM - Sun, 22 May 2022 13:24 UTC

William schrieb am Samstag, 21. Mai 2022 um 23:47:58 UTC+2:
> On Saturday, May 21, 2022 at 5:37:47 PM UTC-3, WM wrote:
>
>
> > How obvious a contradiction has to result from an additional axiom in order to reject it?
> It does not have to be obvious. It must be a contradiction. The Axiom of Infinity does not lead to a contradiction

You know this because you would not accept a contradiction. Many people differ in that point, for example here:

If the fractions m/n are enumerated by the natural numbers k according to Cantor's function
k = (m + n - 1)(m + n - 2)/2 + m
then all the fractions of the sequence
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
are enumerated.

But if the natural numbers first are in bijection with the integer fractions of the first column of the matrix

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
5/1, 5/2, 5/3, 5/4, ...
....

then they must be distributed over the matrix such that no fraction remains without index. That means, there is a permutation such that the X of the first column

XOOOO...
XOOOO...
XOOOO...
XOOOO...
XOOOO...
....

by being exchanged with the O's cover all matrix positions. All O's will vanish. This is obviously impossible because exchanging cannot reduce them. The number of not indexed fractions, represented by O's, will remain constant forever, in infinity.

Why do you not accept this fact?

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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 by: WM - Sun, 22 May 2022 13:27 UTC

horand....@gmail.com schrieb am Sonntag, 22. Mai 2022 um 02:13:32 UTC+2:

> Simply stating that you *THINK* there is a contradiction and presenting one of your proofs as "evidence" doesn't count.

Meanwhile this proof does count for many:

If the fractions m/n are enumerated by the natural numbers k according to Cantor's function
k = (m + n - 1)(m + n - 2)/2 + m
then all the fractions of the sequence
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
are enumerated.

But if the natural numbers first are in bijection with the integer fractions of the first column of the matrix

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
5/1, 5/2, 5/3, 5/4, ...
....

then they must be distributed over the matrix such that no fraction remains without index. That means, there is a permutation such that the X of the first column

XOOOO...
XOOOO...
XOOOO...
XOOOO...
XOOOO...
....

by being exchanged with the O's cover all matrix positions. All O's will vanish. This is obviously impossible because exchanging cannot reduce them. The number of not indexed fractions, represented by O's, will remain constant forever, in infinity.

Why do you not accept this fact?

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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 by: WM - Sun, 22 May 2022 13:29 UTC

FromTheRafters schrieb am Sonntag, 22. Mai 2022 um 03:26:47 UTC+2:
> WM presented the following explanation :

> >> A bunch of 'would be' elements are not a set.
> >
> > Which bunch of finitely many elements would not be a set?
> Those which are not set-like. Proper Classes for instance.

Do you know an example of a finite collection which is not a set?

REgards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
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 by: sergi o - Sun, 22 May 2022 14:47 UTC

On 5/22/2022 8:29 AM, WM wrote:
> FromTheRafters schrieb am Sonntag, 22. Mai 2022 um 03:26:47 UTC+2:
>> WM presented the following explanation :
>
>>>> A bunch of 'would be' elements are not a set.
>>>
>>> Which bunch of finitely many elements would not be a set?
>> Those which are not set-like. Proper Classes for instance.
>
> Do you know an example of a finite collection which is not a set?
>
> REgards, WM

sure
here is a finite grouping of characters on one line;
1 2 3 a

to make them a set in Math, I have to declare it as such and use the notation

this is the set of 1,2,3,a
{1,2,3,a}

=> collections are outside of Math and are not sets.

Re: Natural numbers and vases

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 by: sergi o - Sun, 22 May 2022 14:50 UTC

On 5/22/2022 8:24 AM, WM wrote:
> William schrieb am Samstag, 21. Mai 2022 um 23:47:58 UTC+2:
>> On Saturday, May 21, 2022 at 5:37:47 PM UTC-3, WM wrote:
>>
>>
>>> How obvious a contradiction has to result from an additional axiom in order to reject it?
>> It does not have to be obvious. It must be a contradiction. The Axiom of Infinity does not lead to a contradiction
>
> You know this because you would not accept a contradiction. Many people differ in that point, for example here:
>
> If the fractions m/n are enumerated by the natural numbers k according to Cantor's function
> k = (m + n - 1)(m + n - 2)/2 + m
> then all the fractions of the sequence
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
> are enumerated.

>
> But if the natural numbers first are in bijection with the integer fractions of the first column of the matrix
>
> 1/1, 1/2, 1/3, 1/4, ...
> 2/1, 2/2, 2/3, 2/4, ...
> 3/1, 3/2, 3/3, 3/4, ...
> 4/1, 4/2, 4/3, 4/4, ...
> 5/1, 5/2, 5/3, 5/4, ...
> ...

here is where you make your mistake

>
> then they must be distributed over the matrix such that no fraction remains without index. That means, there is a permutation such that the X of the first column
>
> XOOOO...
> XOOOO...
> XOOOO...
> XOOOO...
> XOOOO...
> ...
>
> by being exchanged with the O's cover all matrix positions. All O's will vanish. This is obviously impossible because exchanging cannot reduce them. The number of not indexed fractions, represented by O's, will remain constant forever, in infinity.
>
> Why do you not accept this fact?

It is wrong, your reasoning is flawed, and you know this.

>
> Regards, WM

Re: Natural numbers and vases

<t6diq3$mve$1@gioia.aioe.org>

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https://www.novabbs.com/tech/article-flat.php?id=100724&group=sci.math#100724

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From: inva...@invalid.com (sergi o)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Sun, 22 May 2022 09:53:54 -0500
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 by: sergi o - Sun, 22 May 2022 14:53 UTC

On 5/22/2022 8:20 AM, WM wrote:
> FromTheRafters schrieb am Sonntag, 22. Mai 2022 um 03:38:11 UTC+2:
>> WM formulated on Saturday :
>>> William schrieb am Samstag, 21. Mai 2022 um 18:08:14 UTC+2:
>>>> On Saturday, May 21, 2022 at 1:00:38 PM UTC-3, WM wrote:
>>>>> William schrieb am Samstag, 21. Mai 2022 um 09:22:27 UTC+2:
>>>>>> On Friday, May 20, 2022 at 8:07:53 AM UTC-3, WM wrote:
>>>>>>> ... not any has less means all together have not less.
>>>>>>
>>>>>> Nope. Trivial counterexample Consider a Peano set S. Every element of S
>>>>>> has a successor in S. S does not have a successor in S.
>>>>> There is no set S
>>>> Piffle. S exists by the Axiom of infinity, It is well known that you do not
>>>> like the results of taking this axiom, but results that you do no like are
>>>> not contradictions.
>>>
>>> The square circle exists by a similarly invalid axiom. Here are some more:
>>>  Axiom of two even primes: There are two even prime numbers.
>>>  Axiom of prime number triples: There is a second triple of prime numbers,
>>> besides (3, 5, 7).  Axiom of meagre sum: There is a set of n different
>>> positive natural numbers with sum n2/2. How obvious a contradiction has to
>>> result from an additional axiom in order to reject it?
>> What is an axiom again?
>
> Depends. A good axiom is a brief statement describing well-established fact.

Wrong. An Axiom is a mathematical statement that is assumed to be true.

you cannot even get the basics right

>
> Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: franz.fr...@gmail.com (Fritz Feldhase)
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 by: Fritz Feldhase - Sun, 22 May 2022 15:00 UTC

On Sunday, May 22, 2022 at 3:29:12 PM UTC+2, WM wrote:
> FromTheRafters schrieb am Sonntag, 22. Mai 2022 um 03:26:47 UTC+2:
> > WM presented the following explanation :
> > > >
> > > > A bunch of 'would be' elements are not a set.
> > > >
> > > Which bunch of finitely many ['would be'] elements would not be a set?
> > >
> > Those which are not set-like. Proper Classes for instance.
> >
> Do you know an example of a finite [bunch of 'would be' elements] which is not a set?

Sure, consider the two proper classes ORD (the class of all ordinal numbers) and R (the Russell class).

These two (proper) classes do not form a set. I.e. There's no set X such that Ax(x e X <-> x = ORD v x = R).*)

You are dumb like shit, Mücke!

__________________________________________________________

*) At least not in the common set theories NBG or MK which allow for "proper classes" besides sets.
See: https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory
and: https://en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theory

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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 by: William - Sun, 22 May 2022 15:07 UTC

On Sunday, May 22, 2022 at 10:24:53 AM UTC-3, WM wrote:
> All O's will vanish. This is obviously impossible

Piffle. The intersection of a sequence of sets, each of cardinality alpeh_0, can be empty. This is a simple result of the Axiom of Infinity, It is a result that you do not like, not a contradiction.

--
William Hughes

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: franz.fr...@gmail.com (Fritz Feldhase)
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 by: Fritz Feldhase - Sun, 22 May 2022 15:09 UTC

On Sunday, May 22, 2022 at 3:24:53 PM UTC+2, WM wrote:

> If the fractions m/n are enumerated by the natural numbers k according to Cantor's function
> k = (m + n - 1)(m + n - 2)/2 + m
> then all the fractions of the sequence
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
> are enumerated.
>
> [I]f the natural numbers [...] are in bijection with the integer fractions of the first column of the matrix
>
> 1/1, 1/2, 1/3, 1/4, ...
> 2/1, 2/2, 2/3, 2/4, ...
> 3/1, 3/2, 3/3, 3/4, ...
> 4/1, 4/2, 4/3, 4/4, ...
> 5/1, 5/2, 5/3, 5/4, ...
> ...
>
> then they [can] be distributed over the matrix such that no fraction remains without index. That means, there is a [distribution] such that the X of the first column
>
> XOOOO...
> XOOOO...
> XOOOO...
> XOOOO...
> XOOOO...
> ...
>
> [...] cover all matrix positions.

Indeed!

There's a simple distribution of the X to achieve this goal:

> 1/1, 1/2, 1/4, 1/7, ...
> 1/3, 1/5, 1/8, ...
> 1/6, 1/9, ...
> 1/10, ...
> ...

> Why do you not accept this fact?

No one knows.

Re: Natural numbers and vases

<t6dn6g$g5d$1@dont-email.me>

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From: nom...@afraid.org (FromTheRafters)
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Subject: Re: Natural numbers and vases
Date: Sun, 22 May 2022 09:08:46 -0700
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 by: FromTheRafters - Sun, 22 May 2022 16:08 UTC

WM used his keyboard to write :
> FromTheRafters schrieb am Sonntag, 22. Mai 2022 um 03:26:47 UTC+2:
>> WM presented the following explanation :
>
>>>> A bunch of 'would be' elements are not a set.
>>>
>>> Which bunch of finitely many elements would not be a set?
>> Those which are not set-like. Proper Classes for instance.
>
> Do you know an example of a finite collection which is not a set?

The best non-fiction writers.

Re: Natural numbers and vases

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From: nom...@afraid.org (FromTheRafters)
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Subject: Re: Natural numbers and vases
Date: Sun, 22 May 2022 09:15:00 -0700
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 by: FromTheRafters - Sun, 22 May 2022 16:15 UTC

WM explained on 5/22/2022 :
> FromTheRafters schrieb am Sonntag, 22. Mai 2022 um 03:38:11 UTC+2:
>> WM formulated on Saturday :
>>> William schrieb am Samstag, 21. Mai 2022 um 18:08:14 UTC+2:
>>>> On Saturday, May 21, 2022 at 1:00:38 PM UTC-3, WM wrote:
>>>>> William schrieb am Samstag, 21. Mai 2022 um 09:22:27 UTC+2:
>>>>>> On Friday, May 20, 2022 at 8:07:53 AM UTC-3, WM wrote:
>>>>>>> ... not any has less means all together have not less.
>>>>>>
>>>>>> Nope. Trivial counterexample Consider a Peano set S. Every element of S
>>>>>> has a successor in S. S does not have a successor in S.
>>>>> There is no set S
>>>> Piffle. S exists by the Axiom of infinity, It is well known that you do
>>>> not like the results of taking this axiom, but results that you do no
>>>> like are not contradictions.
>>>
>>> The square circle exists by a similarly invalid axiom. Here are some more:
>>>  Axiom of two even primes: There are two even prime numbers.
>>>  Axiom of prime number triples: There is a second triple of prime numbers,
>>> besides (3, 5, 7).  Axiom of meagre sum: There is a set of n different
>>> positive natural numbers with sum n2/2. How obvious a contradiction has to
>>> result from an additional axiom in order to reject it?
>> What is an axiom again?
>
> Depends. A good axiom is a brief statement describing well-established fact.
> Example: Group axioms. A bad axiom is arbitrary nonsense, often contradicting
> well-established fact. Example: Large cardinal axioms.

Okay, so what is a theorem then?

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: franz.fr...@gmail.com (Fritz Feldhase)
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 by: Fritz Feldhase - Sun, 22 May 2022 16:41 UTC

On Friday, May 20, 2022 at 11:34:19 PM UTC+2, Fritz Feldhase wrote:
> On Friday, May 20, 2022 at 1:07:53 PM UTC+2, WM wrote:
> >
> > [A] set is nothing else but all its elements.
> >
> Nonsense. [...]

Look, dumbo, if this were true then the set {A} would have to be A, which is NOT the case.

Hint: If A = {0, 1} then A has exactly two elements, namely 0 and 1. But {A} has only o n e element, namely A.

Hint: forall x(x e {A} <-> x = A), hence E!x(x e {A}).

You are mixing up set theory with mereology, you silly crank.

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: franz.fr...@gmail.com (Fritz Feldhase)
Injection-Date: Sun, 22 May 2022 17:04:47 +0000
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 by: Fritz Feldhase - Sun, 22 May 2022 17:04 UTC

On Sunday, May 22, 2022 at 5:09:11 PM UTC+2, Fritz Feldhase wrote:
> On Sunday, May 22, 2022 at 3:24:53 PM UTC+2, WM wrote:
>
> > If the fractions m/n are enumerated by the natural numbers k according to Cantor's function
> > k = (m + n - 1)(m + n - 2)/2 + m
> > then all the fractions of the sequence
> > 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
> > are enumerated.
> >
> > [I]f the natural numbers [...] are in bijection with the integer fractions of the first column of the matrix
> >
> > 1/1, 1/2, 1/3, 1/4, ...
> > 2/1, 2/2, 2/3, 2/4, ...
> > 3/1, 3/2, 3/3, 3/4, ...
> > 4/1, 4/2, 4/3, 4/4, ...
> > 5/1, 5/2, 5/3, 5/4, ...
> > ...
> >
> > then they [can] be distributed over the matrix such that no fraction remains without index. That means, there is a [distribution] such that the X of the first column
> >
> > XOOOO...
> > XOOOO...
> > XOOOO...
> > XOOOO...
> > XOOOO...
> > ...
> >
> > [...] cover all matrix positions.
>
> Indeed!
>
> There's a simple distribution of the X to achieve this goal

namely:

> > 1/1, 2/1, 4/1, 7/1, ...
> > 3/1, 5/1, 8/1, ...
> > 6/1, 9/1, ...
> > 10/1, ...
> > ...

> > Why do you not accept this fact?
> >
> No one knows.

Re: Natural numbers and vases

<t6e08d$pdd$1@gioia.aioe.org>

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From: iel...@didellee.ei (Dell Ibi)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
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 by: Dell Ibi - Sun, 22 May 2022 18:43 UTC

FromTheRafters wrote:

>> Depends. A good axiom is a brief statement describing well-established
>> fact. Example: Group axioms. A bad axiom is arbitrary nonsense, often
>> contradicting well-established fact. Example: Large cardinal axioms.
>
> Okay, so what is a theorem then?

your country is a gas station mascaraed as a country.

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
Injection-Date: Mon, 23 May 2022 11:25:17 +0000
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 by: WM - Mon, 23 May 2022 11:25 UTC

Fritz Feldhase schrieb am Sonntag, 22. Mai 2022 um 17:00:16 UTC+2:
> On Sunday, May 22, 2022 at 3:29:12 PM UTC+2, WM wrote:

> > Do you know an example of a finite [bunch of 'would be' elements] which is not a set?
>
> Sure, consider the two proper classes ORD (the class of all ordinal numbers) and R (the Russell class).

They are a set of two elements.
>
> These two (proper) classes do not form a set. I.e. There's no set X such that Ax(x e X <-> x = ORD v x = R).*)

An element of a set need not consist of elements.

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
Injection-Date: Mon, 23 May 2022 11:30:59 +0000
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 by: WM - Mon, 23 May 2022 11:30 UTC

William schrieb am Sonntag, 22. Mai 2022 um 17:07:48 UTC+2:
> On Sunday, May 22, 2022 at 10:24:53 AM UTC-3, WM wrote:
> > All O's will vanish. This is obviously impossible
> Piffle. The intersection of a sequence of sets, each of cardinality alpeh_0, can be empty.

Here we are not at all interested in any intersection but whether there is a state where all O's have disappeared. This is not the case because the share of O's does never change. Your claim would require that the sequence -1 + 1 - 1 + 1 - 1 + 1 -+ = oo.

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
Injection-Date: Mon, 23 May 2022 11:37:40 +0000
Content-Type: text/plain; charset="UTF-8"
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 by: WM - Mon, 23 May 2022 11:37 UTC

Fritz Feldhase schrieb am Sonntag, 22. Mai 2022 um 19:04:52 UTC+2:
> On Sunday, May 22, 2022 at 5:09:11 PM UTC+2, Fritz Feldhase wrote:
> > On Sunday, May 22, 2022 at 3:24:53 PM UTC+2, WM wrote:
> >
> > > If the fractions m/n are enumerated by the natural numbers k according to Cantor's function
> > > k = (m + n - 1)(m + n - 2)/2 + m
> > > then all the fractions of the sequence
> > > 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
> > > are enumerated.
> > >
> > > [I]f the natural numbers [...] are in bijection with the integer fractions of the first column of the matrix
> > >
> > > 1/1, 1/2, 1/3, 1/4, ...
> > > 2/1, 2/2, 2/3, 2/4, ...
> > > 3/1, 3/2, 3/3, 3/4, ...
> > > 4/1, 4/2, 4/3, 4/4, ...
> > > 5/1, 5/2, 5/3, 5/4, ...
> > > ...
> > >
> > > then they [can] be distributed over the matrix such that no fraction remains without index. That means, there is a [distribution] such that the X of the first column
> > >
> > > XOOOO...
> > > XOOOO...
> > > XOOOO...
> > > XOOOO...
> > > XOOOO...
> > > ...
> > >
> > > [...] cover all matrix positions.
> >
> > Indeed!
> >
> > There's a simple distribution of the X to achieve this goal
> namely:
>
> > > 1/1, 2/1, 4/1, 7/1, ...
> > > 3/1, 5/1, 8/1, ...
> > > 6/1, 9/1, ...
> > > 10/1, ...
> > > ...
>
That appears so to you because you don't know dark numbers. But they are proved by the fact that the share of enumerated fractions does never change. Your claim would require that the series -1 + 1 - 1 + 1 - 1 + 1 -+ = oo.

Regards, WM

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