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Obviously I was either onto something, or on something. -- Larry Wall on the creation of Perl


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

<t5getf$1dug$2@gioia.aioe.org>

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https://www.novabbs.com/tech/article-flat.php?id=99688&group=sci.math#99688

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Path: i2pn2.org!i2pn.org!aioe.org!jq9Zon5wYWPEc6MdU7JpBw.user.46.165.242.75.POSTED!not-for-mail
From: inva...@invalid.com (sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Wed, 11 May 2022 08:49:33 -0500
Organization: Aioe.org NNTP Server
Message-ID: <t5getf$1dug$2@gioia.aioe.org>
References: <06d0f903-3235-4be2-8653-85563102704fn@googlegroups.com>
<fcd32c40-18c2-457d-8f71-e2d378bf3288n@googlegroups.com>
<756a88cf-a3e6-4ad9-93dc-627eef357e8dn@googlegroups.com>
<9756fced-2bdf-4272-8801-b2654392c1f1n@googlegroups.com>
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 by: sergio - Wed, 11 May 2022 13:49 UTC

On 5/11/2022 7:52 AM, WM wrote:
> FromTheRafters schrieb am Dienstag, 10. Mai 2022 um 23:14:30 UTC+2:
>> WM presented the following explanation :
>>> William schrieb am Dienstag, 10. Mai 2022 um 22:44:07 UTC+2:
>>>> On Tuesday, May 10, 2022 at 5:34:38 PM UTC-3, WM wrote:
>>>
>>>>> What I meant is this: Divide the set 1, 2, 3, ... at any point you like.
>>>>> The lower set is finite.
>>>> And the upper set is infinite, You can do this at any point you like, so you
>>>> can do this at each of an infinite number of elements.
>>>
>>> No. Always an infinite number must remain in the upper set. But it is
>>> impossible to divide 1, 2, 3, ... into two infinite sets.
>> I can collect the odds and the evens into two infinite sets.
>
> But here we are talking about consecutive sets: lower and upper set.
> Divide the set 1, 2, 3, ... at any point you like.

try using Math to explain yourself.

>
> Regards, WM

Re: Natural numbers and vases

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From: inva...@invalid.com (sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Wed, 11 May 2022 08:54:27 -0500
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 by: sergio - Wed, 11 May 2022 13:54 UTC

On 5/11/2022 7:59 AM, WM wrote:
> Jim Burns schrieb am Mittwoch, 11. Mai 2022 um 02:57:03 UTC+2:
>> On 5/10/2022 4:46 PM, WM wrote:
>
>>>>> All fractions to be indexed are in this 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, ...
>>> ...
>
>> Assign the index m+(m+n-1)*(m+n-2)/2
>> to the fraction m/n
>
> That is done. All assigned fractions are gathered in the first column.
>>
>> The number of fractions not-indexed is zero.
>
> Wrong. The number of fractions never in the first column is infinite.

Wrong. k = m+(m+n-1)*(m+n-2)/2

for all k, m, n at all times.

too bad you do not understand equations.

But you have to give up Truth to claim Cantor was wrong.

>> Initially,
>> k = m+(m+n-1)*(m+n-2)/2
>>
>> So, no.
>
> Neither k nor m nor n is a natural number.

Wrong, they are all natural numbers.

you don't know about classes of numbers, types of numbers, or units either.

> Natural numbers can be inserted for these variables. The outset matrix above contains many fractions not in the first column. The k are the indices of the first column: 11, 21, 31, ... .

nonsense.

>
> Regards, WM
>
>

Re: Natural numbers and vases

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From: inva...@invalid.com (sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Wed, 11 May 2022 08:56:17 -0500
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 by: sergio - Wed, 11 May 2022 13:56 UTC

On 5/11/2022 8:15 AM, WM wrote:
> Fritz Feldhase schrieb am Mittwoch, 11. Mai 2022 um 03:40:28 UTC+2:
>> On Tuesday, May 10, 2022 at 10:46:57 UTC+2, WM wrote:
>>>>
>>>> Assign the index m+(m+n-1)*(m+n-2)/2 to the fraction m/n. [Jim Beam]
>>>>
>>> there are infinitely many not indexed fractions
>> Nope. If q is a (any) fraction, then there are two unique natural numbers n, m such that q is n/m (since that is what the notion "fraction" means) and its index is m+(m+n-1)*(m+n-2)/2.
>>
>> Hence "The number of fractions not-indexed is zero".
>
> Unfortunately too short-sighted. Of course all fractions of Cantor's sequence 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ... are indexed as they can be found in the first column. Nevertheless all other columns are populated too. Do you agree?
>
> Regards, WM

show which fraction is not indexed. All are indexed.

"The number of fractions not-indexed is zero"

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: Wed, 11 May 2022 09:09:15 -0700
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 by: FromTheRafters - Wed, 11 May 2022 16:09 UTC

WM presented the following explanation :
> FromTheRafters schrieb am Dienstag, 10. Mai 2022 um 23:14:30 UTC+2:
>> WM presented the following explanation :
>>> William schrieb am Dienstag, 10. Mai 2022 um 22:44:07 UTC+2:
>>>> On Tuesday, May 10, 2022 at 5:34:38 PM UTC-3, WM wrote:
>>>>> What I meant is this: Divide the set 1, 2, 3, ... at any point you like.
>>>>> The lower set is finite.
>>>> And the upper set is infinite, You can do this at any point you like, so
>>>> you can do this at each of an infinite number of elements.
>>>
>>> No. Always an infinite number must remain in the upper set. But it is
>>> impossible to divide 1, 2, 3, ... into two infinite sets.
>> I can collect the odds and the evens into two infinite sets.
>
> But here we are talking about consecutive sets: lower and upper set.
> Divide the set 1, 2, 3, ... at any point you like.

One set with the FISON and its complement considered separately. Yes, I
know what you meant. Partitioning. It is well known how that works with
this order type.

Re: Natural numbers and vases

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Wed, 11 May 2022 15:20:51 -0400
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 by: Jim Burns - Wed, 11 May 2022 19:20 UTC

On 5/11/2022 8:59 AM, WM wrote:
> Jim Burns schrieb
> am Mittwoch, 11. Mai 2022 um 02:57:03 UTC+2:
>> On 5/10/2022 4:46 PM, WM wrote:

>>>>> All fractions to be indexed are in this 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, ...
>>> ...

For each fraction m/n in the matrix,

the column above m/n is a finite sequence
⟨ 1/n, 2/n, ..., m/n ⟩ such that,
for each split BEFORE and AFTER,
BEFORE ends with j/n and
AFTER begins with (j+1)/n,

and
the row left of m/n is a finite sequence
⟨ m/1, m/2, ..., m/n ⟩ such that,
for each split BEFORE and AFTER,
BEFORE ends with m/i and
AFTER begins with m/(i+1).

That's true of each fraction in the matrix.

Whatever further claim we can reason to
starting from this claim about m/n
is also true of each fraction in the matrix.

>> Assign the index m+(m+n-1)*(m+n-2)/2
>> to the fraction m/n
>
> That is done.
> All assigned fractions are gathered in the first column.

We know that,
for each fraction m/n in the matrix,

the column above (m+(m+n-1)*(m+n-2)/2)/1 is
a finite sequence
⟨ 1/1, 2/1, ..., (m+(m+n-1)*(m+n-2)/2)/1 ⟩
such that, for each split BEFORE and AFTER,
BEFORE ends with j/1 and
AFTER begins with (j+1)/1,

We know this by starting from a claim true of
each fraction in the matrix, and
reasoning from that claim, which thus must be true
of each fraction in the matrix.

>> The number of fractions not-indexed is zero.
>
> Wrong.
> The number of fractions never in the first column
> is infinite.

.... is zero.

Each m/n is at (m+(m+n-1)*(m+n-2)/2)/1

> Allmatrix positions remain occupied even if
> all definable fractions 1/1, 1/2, 2/1, 1/3, 2/2, 3/1,
> 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1,
> 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ...
> are gathered in the first column.

"All matrix positions remain occupied" follows from
the assumption that all matrix positions
have Bob-conservation --
"Bob" being the name of the O which starts at
the position of 1/2 in the matrix, and then
swaps positions with X's at 2/1, 3/1, 6/1, ...

https://www.youtube.com/watch?v=TjAg-8qqR3g

What you have before you,
in which each m/n is at (m+(m+n-1)*(m+n-2)/2)/1,
is proof that the collection of all matrix positions
*does not* have Bob-conservation.

Bob is in the matrix.
No step which Bob takes leaves Bob outside the matrix.
After all the steps, Bob is nowhere in the matrix.

Before: there is Bob.
After: Bob is not outside and Bob is not inside.
Therefore, Bob is not conserved.

Each FISON conserves Bob.

The union of all FISONs does not conserve Bob.

>>> So it appears to the naive mind.
>>> But there are infinitely many not indexed fractions
>>> initially
>>
>> Initially,
>> k = m+(m+n-1)*(m+n-2)/2
>>
>> So, no.
>
> Neither k nor m nor n is a natural number.
> Natural numbers can be inserted for these variables.

Variables are the mathematical equivalent of
pronouns.

If I say "It is a horse", that could be true,
that could be false, depending upon the context
of where, when, how I say it.

When "It is a horse" is true, "it" _refers to_
a horse. It is true that "it", the pronoun,
is never a horse. No one ever thinks it ever is.

When "k is a natural number" is true,
k _refers to_ a natural number.
It is true that 'k', the variable, is never a
natural number. No one ever thinks it ever is.

> The outset matrix above contains many fractions
> not in the first column. The k are the indices of
> the first column: 11, 21, 31, ... .

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: franz.fr...@gmail.com (Fritz Feldhase)
Injection-Date: Wed, 11 May 2022 22:46:13 +0000
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 by: Fritz Feldhase - Wed, 11 May 2022 22:46 UTC

On Wednesday, May 11, 2022 at 2:30:52 PM UTC+2, horand....@gmail.com wrote:
> On Tuesday, 10 May 2022 at 22:40:28 UTC-3, Fritz Feldhase wrote:
> > On Tuesday, May 10, 2022 at 10:46:57 UTC+2, WM wrote:
>
> [nonsense]

:-)

> > Nope. If q is a (any) fraction, then there are two unique natural numbers n, m such that q is n/m (since that is what the notion "fraction" means) and its index is m+(m+n-1)*(m+n-2)/2.
> >
> That's not quite true. [...]

Quite true. Note that a "fraction" in the context of this discussion does not denote an element in Q but an element in {(n, m) : n, m e IN}. Moreover, n/m := (n, m) with n, m e IN.

No really, _in the present context_ (referring to "fractions") we have, say, 2/3 =/= 3/6.

Are we clear?

So if q is a "fraction", i. e. q e {(n, m) : n, m e IN}, "then there are two unique natural numbers n, m such that q is n/m", i. e. q = (n, m). No? :-P

Again, 2/3 = (2, 3) =/= (3, 6) = 3/6.

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 - Thu, 12 May 2022 00:42 UTC

> ... In order to have an infinite set, you need to take all

So what? My claim is not that there is

(i) "an infinite set of elements of N_p [Call the set B] which is followed by an infinite set of elements of N_p"

but that there is

(ii) an infinite set of elements of N_p, call the set B, *each element of* which is followed by an infinite set of elements of N_p.

call the element n and a set the works for n, S(n) [there is no set S which works for every n]. You can not conclude from this that there is an infinite set of elements of N_p that follows B, that is (ii) does note impy (i).

--
William Hughes

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
Injection-Date: Thu, 12 May 2022 13:06:24 +0000
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 by: WM - Thu, 12 May 2022 13:06 UTC

Jim Burns schrieb am Mittwoch, 11. Mai 2022 um 21:21:03 UTC+2:
> On 5/11/2022 8:59 AM, WM wrote:

> > The number of fractions never in the first column
> > is infinite.
> ... is zero.
>
> Each m/n is at (m+(m+n-1)*(m+n-2)/2)/1

Each m/n so defined has aleph_0 successors. Therefore your argument is void.

> > Allmatrix positions remain occupied even if
> > all definable fractions 1/1, 1/2, 2/1, 1/3, 2/2, 3/1,
> > 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1,
> > 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ...
> > are gathered in the first column.
> "All matrix positions remain occupied" follows from
> the assumption

No it follows from the fact that only exchanges take place which never can empty a place.

> What you have before you,
> in which each m/n is at (m+(m+n-1)*(m+n-2)/2)/1,
> is proof that the collection of all matrix positions
> *does not* have Bob-conservation.
>
> Bob is in the matrix.
> No step which Bob takes leaves Bob outside the matrix.
> After all the steps, Bob is nowhere in the matrix.

Wrong. It is impossible that something leaves the matrix. But there are dark positions.

> > Neither k nor m nor n is a natural number.
> > Natural numbers can be inserted for these variables.

> When "k is a natural number" is true,
> k _refers to_ a natural number.

It is not true. k can be replaced by a natural number.

> It is true that 'k', the variable, is never a
> natural number. No one ever thinks it ever is.

Fine. Therefore your formula k = m+(m+n-1)*(m+n-2)/2 does not prove something for all numbers.

> > The outset matrix above contains many fractions
> > not in the first column. The k are the indices of
> > the first column: 11, 21, 31, ... .

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 - Thu, 12 May 2022 13:07 UTC

William schrieb am Donnerstag, 12. Mai 2022 um 02:42:39 UTC+2:
> > ... In order to have an infinite set, you need to take all
>
> So what? My claim is not that there is
>
> (i) "an infinite set of elements of N_p [Call the set B] which is followed by an infinite set of elements of N_p"

Fine. That would be easily disproved.
>
> but that there is
>
> (ii) an infinite set of elements of N_p, call the set B, *each element of* which is followed by an infinite set of elements of N_p.

Each element which is followed by an infinite set belongs to a finite set, a FISON, and is provably not required to make the set ℕ infinite. Therefore you can remove it.
∀n ∈ ℕ_p: |ℕ \ {1, 2, 3, ..., n}| = ℵo

> call the element n and a set the works for n, S(n) [there is no set S which works for every n].

Every element having an infinite set S(n) of successors can be removed. What remains? If nothing remains, then not every element could have infinitely many successors. If something remains, then it is dark.

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
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 by: sergio - Thu, 12 May 2022 15:14 UTC

On 5/12/2022 8:06 AM, WM wrote:
> Jim Burns schrieb am Mittwoch, 11. Mai 2022 um 21:21:03 UTC+2:
>> On 5/11/2022 8:59 AM, WM wrote:
>
>>> The number of fractions never in the first column
>>> is infinite.
>> ... is zero.
>>
>> Each m/n is at (m+(m+n-1)*(m+n-2)/2)/1
>
> Each m/n so defined has aleph_0 successors. Therefore your argument is void.

Whoa!!

>
>>> Allmatrix positions remain occupied even if
>>> all definable fractions 1/1, 1/2, 2/1, 1/3, 2/2, 3/1,
>>> 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1,
>>> 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ...
>>> are gathered in the first column.
>> "All matrix positions remain occupied" follows from
>> the assumption
>
> No it follows from the fact that only exchanges take place which never can empty a place.

that is not fact at all. Your goofy exchanges are deception.

>
>> What you have before you,
>> in which each m/n is at (m+(m+n-1)*(m+n-2)/2)/1,
>> is proof that the collection of all matrix positions
>> *does not* have Bob-conservation.
>>
>> Bob is in the matrix.
>> No step which Bob takes leaves Bob outside the matrix.
>> After all the steps, Bob is nowhere in the matrix.
>
> Wrong. It is impossible that something leaves the matrix. But there are dark positions.

more lies.

>
>>> Neither k nor m nor n is a natural number.
>>> Natural numbers can be inserted for these variables.
>
>> When "k is a natural number" is true,
>> k _refers to_ a natural number.
>
> It is not true. k can be replaced by a natural number.

Wrong. k represents a natural number.

you never took Algebra.

>
>> It is true that 'k', the variable, is never a
>> natural number. No one ever thinks it ever is.
>
> Fine. Therefore your formula k = m+(m+n-1)*(m+n-2)/2 does not prove something for all numbers.

no. k = m+(m+n-1)*(m+n-2)/2 is mapping of all rational numbers to all natural numbers, no exceptions and is good at all times.

>
>>> The outset matrix above contains many fractions
>>> not in the first column. The k are the indices of
>>> the first column: 11, 21, 31, ... .
>
> Regards, WM

Re: Natural numbers and vases

<88315a36-8579-84b5-2b44-82ceec0461f0@att.net>

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Thu, 12 May 2022 13:05:28 -0400
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 by: Jim Burns - Thu, 12 May 2022 17:05 UTC

On 5/12/2022 9:06 AM, WM wrote:
> Jim Burns schrieb
> am Mittwoch, 11. Mai 2022 um 21:21:03 UTC+2:
>> On 5/11/2022 8:59 AM, WM wrote:

>>> Allmatrix positions remain occupied even if
>>> all definable fractions 1/1, 1/2, 2/1, 1/3, 2/2, 3/1,
>>> 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1,
>>> 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ...
>>> are gathered in the first column.
>>
>> "All matrix positions remain occupied" follows from
>> the assumption

that all matrix positions
have Bob-conservation --
"Bob" being the name of the O which starts at
the position of 1/2 in the matrix, and then
swaps positions with X's at 2/1, 3/1, 6/1, ...

https://www.youtube.com/watch?v=TjAg-8qqR3g

> No it follows from the fact that
> only exchanges take place which never can empty a place.

Start Bob at 1.
OXXXXX...

Exchange 1 and 2. No empty place.
XOXXXX...

Exchange 2 and 3. No empty place.
XXOXXX...

Exchange 3 and 4. No empty place.
XXXOXX...

....

Bob never exchanges to any place which
cannot (by some swap-sequence) exchange with 1.

Do all the exchanges. No Bob.
XXXXXX...

Bob does not go to any place which
cannot exchange with 1.
Bob is not left at any place which
can exchange with 1.
No Bob. Anywhere.

In order for Bob to be somewhere after
all the exchanges, there would need to be
some place which Bob can exchange into and
not out of.

Dark places, existing or not, have no effect
because they are not places Bob can exchange into.

Describe a place which Bob can exchange into.[1]
From that description, we will know that
it is also a place Bob can exchange out of.

Therefore,
there is no place which Bob can exchange into
and not out of.

Therefore,
Bob is _not_ somewhere after all the exchanges.
No Bob-conservation.

----
[1]
Describe a place which Bob can exchange into.

k is a place which Bob can exchange into iff
each non-empty collection =< k
contains a first and a last,
for each split BEFORE and AFTER of positions =< k.
i is last in BEFORE, j is first in AFTER,
and j = i+1

Assume that k is a place which Bob can exchange into.
Append k+1 after all the places =< k
Work through the description of
a place which Bob can exchange into.
k+1 satisfies that description.
k+1 is also a place which Bob can exchange into.

Therefore,
no place which Bob can exchange into is
a place which Bob cannot exchange out of.

Therefore,
no Bob-conservation.

>> What you have before you,
>> in which each m/n is at (m+(m+n-1)*(m+n-2)/2)/1,
>> is proof that the collection of all matrix positions
>> *does not* have Bob-conservation.
>>
>> Bob is in the matrix.
>> No step which Bob takes leaves Bob outside the matrix.
>> After all the steps, Bob is nowhere in the matrix.
>
> Wrong.
> It is impossible that something leaves the matrix.
> But there are dark positions.

Dark places, existing or not, have no effect
because they are not places Bob can exchange into.

Re: Natural numbers and vases

<e079b167-337d-2807-d77e-cded60ade916@att.net>

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From: james.g....@att.net (Jim Burns)
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Subject: Re: Natural numbers and vases
Date: Thu, 12 May 2022 13:35:40 -0400
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 by: Jim Burns - Thu, 12 May 2022 17:35 UTC

On 5/12/2022 1:05 PM, Jim Burns wrote:
> On 5/12/2022 9:06 AM, WM wrote:
>> Jim Burns schrieb
>> am Mittwoch, 11. Mai 2022 um 21:21:03 UTC+2:
>>> On 5/11/2022 8:59 AM, WM wrote:

>>>> Allmatrix positions remain occupied even if
>>>> all definable fractions 1/1, 1/2, 2/1, 1/3, 2/2, 3/1,
>>>> 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1,
>>>> 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ...
>>>> are gathered in the first column.
>>>
>>> "All matrix positions remain occupied" follows from
>>> the assumption
>
> that all matrix positions
> have Bob-conservation --
> "Bob" being the name of the O which starts at
> the position of 1/2 in the matrix, and then
> swaps positions with X's at 2/1, 3/1, 6/1, ...
>
> https://www.youtube.com/watch?v=TjAg-8qqR3g
>
>> No it follows from the fact that
>> only exchanges take place which never can empty a place.

tl;dr

> Describe a place which Bob can exchange into.[1]
> From that description, we will know that
> it is also a place Bob can exchange out of.

If
C is an ordered collection which
contains a first and a last,
then
appending k+1 to it == C∪{k+1} yields
an ordered collection C∪{k+1} which
contains a first and a last.

Bob can exchange into places such that
each non-empty collection of places before
contains a first and a last.

Appending one more place yields another place
such that
each non-empty collection of places before
contains a first and a last,
which is another place Bob can exchange into.

Re: Natural numbers and vases

<t5jgoe$1ae0$1@gioia.aioe.org>

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From: inva...@invalid.com (sergio)
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Subject: Re: Natural numbers and vases
Date: Thu, 12 May 2022 12:39:24 -0500
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 by: sergio - Thu, 12 May 2022 17:39 UTC

On 5/12/2022 12:05 PM, Jim Burns wrote:
> On 5/12/2022 9:06 AM, WM wrote:
>> Jim Burns schrieb
>> am Mittwoch, 11. Mai 2022 um 21:21:03 UTC+2:
>>> On 5/11/2022 8:59 AM, WM wrote:
>
>>>> Allmatrix positions remain occupied even if
>>>> all definable fractions 1/1, 1/2, 2/1, 1/3, 2/2, 3/1,
>>>> 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1,
>>>> 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ...
>>>> are gathered in the first column.
>>>
>>> "All matrix positions remain occupied" follows from
>>> the assumption
>
> that all matrix positions
> have Bob-conservation --
> "Bob" being the name of the O which starts at
> the position of 1/2 in the matrix, and then
> swaps positions with X's at 2/1, 3/1, 6/1, ...
>
> https://www.youtube.com/watch?v=TjAg-8qqR3g
>
>> No it follows from the fact that
>> only exchanges take place which never can empty a place.
>
> Start Bob at 1.
> OXXXXX...
>
> Exchange 1 and 2. No empty place.
> XOXXXX...
>
> Exchange 2 and 3. No empty place.
> XXOXXX...
>
> Exchange 3 and 4. No empty place.
> XXXOXX...
>
> ...
>
> Bob never exchanges to any place which
> cannot (by some swap-sequence) exchange with 1.
>
> Do all the exchanges. No Bob.
> XXXXXX...
>
> Bob does not go to any place which
> cannot exchange with 1.
> Bob is not left at any place which
> can exchange with 1.
> No Bob. Anywhere.
>
> In order for Bob to be somewhere after
> all the exchanges, there would need to be
> some place which Bob can exchange into and
> not out of.
>
> Dark places, existing or not, have no effect
> because they are not places Bob can exchange into.
>
> Describe a place which Bob can exchange into.[1]
> From that description, we will know that
> it is also a place Bob can exchange out of.
>
> Therefore,
> there is no place which Bob can exchange into
> and not out of.
>
> Therefore,
> Bob is _not_ somewhere after all the exchanges.
> No Bob-conservation.
>
> ----
> [1]
> Describe a place which Bob can exchange into.
>
> k is a place which Bob can exchange into  iff
> each non-empty collection =< k
> contains a first and a last,
> for each split BEFORE and AFTER of positions =< k.
> i is last in BEFORE, j is first in AFTER,
> and j = i+1
>
> Assume that k is a place which Bob can exchange into.
> Append k+1 after all the places =< k
> Work through the description of
> a place which Bob can exchange into.
> k+1 satisfies that description.
> k+1 is also a place which Bob can exchange into.
>
> Therefore,
> no place which Bob can exchange into is
> a place which Bob cannot exchange out of.
>
> Therefore,
> no Bob-conservation.
>
>>> What you have before you,
>>> in which each m/n is at (m+(m+n-1)*(m+n-2)/2)/1,
>>> is proof that the collection of all matrix positions
>>> *does not* have Bob-conservation.
>>>
>>> Bob is in the matrix.
>>> No step which Bob takes leaves Bob outside the matrix.
>>> After all the steps, Bob is nowhere in the matrix.
>>
>> Wrong.
>> It is impossible that something leaves the matrix.
>> But there are dark positions.
>
> Dark places, existing or not, have no effect
> because they are not places Bob can exchange into.
>

Bob is at = > kbob

kbob = (mbob+(mbob+nbob-1)*(mbob+nbob-2)/2)/1,

so just mbob, nbob me and I send kbob right over

(kbob = King Bob.)

Extra Credit: if mbob is latitude, and nbob is longitiude, would King Bob not be a natural number anymore? would his enumeration with the earths surface
be ok ?

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 - Thu, 12 May 2022 19:23 UTC

On Thursday, May 12, 2022 at 10:07:27 AM UTC-3, WM wrote:
..>..[any single] element having an infinite set S(n) of successors can be removed.

Indeed. Any single *element* of N_p can be removed. (indeed any finite set of elements of N_p). The *set* N_p cannot be removed. There is no single element of N_p (indeed no finite set of elements) which cannot be removed just as there is not single element (indeed no finite set of elements) that makes N_p infinite.

--
William Hughes

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 - Fri, 13 May 2022 17:01 UTC

William schrieb am Donnerstag, 12. Mai 2022 um 21:23:05 UTC+2:
> On Thursday, May 12, 2022 at 10:07:27 AM UTC-3, WM wrote:
> .>..[any single] element having an infinite set S(n) of successors can be removed.
>
> Indeed. Any single *element* of N_p can be removed.

Together with all its predecessors!

> (indeed any finite set of elements of N_p).

There are no other sets defined by natural numbers. Every FISON is finite. No definable natural number is outside of a FISON.

> The *set* N_p cannot be removed.

Which n would resist? Every n that can be counted can be removed. What would be the difference between indexing and removing.

> There is no single element of N_p (indeed no finite set of elements) which cannot be removed just as there is not single element (indeed no finite set of elements) that makes N_p infinite.

That shows that N_p is not actually infinite but only potentially infinite. The only counter argument is your belief.

The set of all FISONs cannot lose all FISONs an remain something else but empty.

Regards, WM

Re: Natural numbers and vases

<t5m3si$17al$1@gioia.aioe.org>

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Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Fri, 13 May 2022 12:18:08 -0500
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 by: sergio - Fri, 13 May 2022 17:18 UTC

On 5/13/2022 12:01 PM, WM wrote:
> William schrieb am Donnerstag, 12. Mai 2022 um 21:23:05 UTC+2:
>> On Thursday, May 12, 2022 at 10:07:27 AM UTC-3, WM wrote:
>> .>..[any single] element having an infinite set S(n) of successors can be removed.
>>
>> Indeed. Any single *element* of N_p can be removed.
>
> Together with all its predecessors!

"remove" is not the right word, as you have changed the set N_p into something else.

>
>> (indeed any finite set of elements of N_p).
>
> There are no other sets defined by natural numbers.

wrong, {17,2,3} is a set defined by natural numbers.

>Every FISON is finite. No definable natural number is outside of a FISON.

there is no such thing as your "definable" number, which requires two people a rapping, tapping flashing lights
>
>> The *set* N_p cannot be removed.
>
> Which n would resist? Every n that can be counted can be removed. What would be the difference between indexing and removing.

again, "removing" is the wrong word

>
>> There is no single element of N_p (indeed no finite set of elements) which cannot be removed just as there is not single element (indeed no finite set of elements) that makes N_p infinite.
>
> That shows that N_p is not actually infinite but only potentially infinite.

Wrong, and outright lie.

> The set of all FISONs cannot lose all FISONs an remain something else but empty.

the set of all FISONs is not a FISON.

>
> Regards, WM
>

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: horand.g...@gmail.com (Gus Gassmann)
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 by: Gus Gassmann - Fri, 13 May 2022 17:18 UTC

On Friday, 13 May 2022 at 14:01:08 UTC-3, WM wrote:
> William schrieb am Donnerstag, 12. Mai 2022 um 21:23:05 UTC+2:
> > On Thursday, May 12, 2022 at 10:07:27 AM UTC-3, WM wrote:
> > .>..[any single] element having an infinite set S(n) of successors can be removed.
> >
> > Indeed. Any single *element* of N_p can be removed.
> Together with all its predecessors!
> > (indeed any finite set of elements of N_p).
> There are no other sets defined by natural numbers. Every FISON is finite. No definable natural number is outside of a FISON.

Shut the fuck up, moron, about things you do not understand. *EVERY* natural number is an element of a FISON, and *EVERY* natural number defines an end segment of cardinality aleph_0.

Re: Natural numbers and vases

<t5mbi0$cck$1@dont-email.me>

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From: nom...@afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Fri, 13 May 2022 12:28:58 -0700
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 by: FromTheRafters - Fri, 13 May 2022 19:28 UTC

WM used his keyboard to write :
> William schrieb am Donnerstag, 12. Mai 2022 um 21:23:05 UTC+2:
>> On Thursday, May 12, 2022 at 10:07:27 AM UTC-3, WM wrote:
>> .>..[any single] element having an infinite set S(n) of successors can be
>> removed.
>>
>> Indeed. Any single *element* of N_p can be removed.
>
> Together with all its predecessors!

That would be a proper subset, not a single element. You could use the
initial element to satisfy this trivially since it has no predecessor.

Again, you are not removing them from a set but only choosing between
removing them, or not removing them, from consideration in building a
new set.

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 - Fri, 13 May 2022 20:25 UTC

FromTheRafters schrieb am Freitag, 13. Mai 2022 um 21:29:14 UTC+2:
> WM used his keyboard to write :
> > William schrieb am Donnerstag, 12. Mai 2022 um 21:23:05 UTC+2:
> >> On Thursday, May 12, 2022 at 10:07:27 AM UTC-3, WM wrote:
> >> .>..[any single] element having an infinite set S(n) of successors can be
> >> removed.
> >>
> >> Indeed. Any single *element* of N_p can be removed.
> >
> > Together with all its predecessors!
> That would be a proper subset, not a single element. You could use the
> initial element to satisfy this trivially since it has no predecessor.
>
> Again, you are not removing them from a set

I am removing elements from a set. A \ B is possible. Cantor does it too: wir könnten sie alsdann von S entfernen, ohne daß diese Menge dadurch die in 2) ausgedrückte Beziehung zu M verliert.

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 - Fri, 13 May 2022 20:27 UTC

horand....@gmail.com schrieb am Freitag, 13. Mai 2022 um 19:18:45 UTC+2:
> On Friday, 13 May 2022 at 14:01:08 UTC-3, WM wrote:
> > William schrieb am Donnerstag, 12. Mai 2022 um 21:23:05 UTC+2:
> > > On Thursday, May 12, 2022 at 10:07:27 AM UTC-3, WM wrote:
> > > .>..[any single] element having an infinite set S(n) of successors can be removed.
> > >
> > > Indeed. Any single *element* of N_p can be removed.
> > Together with all its predecessors!
> > > (indeed any finite set of elements of N_p).
> > There are no other sets defined by natural numbers. Every FISON is finite. No definable natural number is outside of a FISON.
> *EVERY* natural number is an element of a FISON,

No.

> and *EVERY* natural number defines an end segment of cardinality aleph_0.

No.

Regards, WM

Re: Natural numbers and vases

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Fri, 13 May 2022 16:32:05 -0400
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 by: Jim Burns - Fri, 13 May 2022 20:32 UTC

On 5/12/2022 9:06 AM, WM wrote:
> Jim Burns schrieb
> am Mittwoch, 11. Mai 2022 um 21:21:03 UTC+2:
>> On 5/11/2022 8:59 AM, WM wrote:

>>> Allmatrix positions remain occupied even if
>>> all definable fractions 1/1, 1/2, 2/1, 1/3, 2/2, 3/1,
>>> 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1,
>>> 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ...
>>> are gathered in the first column.
>>
>> "All matrix positions remain occupied" follows from
>> the assumption
>
> No it follows from the fact that
> only exchanges take place which never can empty a place.

That seems to be at the core of our discussion.
I expect to be returning to my answer again and again,
so I've cleaned up my writing as best I could.

----
Start Bob at 1.
OXXXXX...

Exchange 1 and 2. No empty place.
XOXXXX...

Exchange 2 and 3. No empty place.
XXOXXX...

Exchange 3 and 4. No empty place.
XXXOXX...

....

No empty places, but
there is no Bob-conservation for all swaps.

Any place that Bob cannot swap _into_
is not where Bob is after all swaps.

Any place that Bob can swap _out of_
is not where Bob is after all swaps.

Describe a place which Bob can swap into.

( k is such a place.
( For each BEFORE and AFTER =< k,
( i ends BEFORE, j begins AFTER, j = i+1

From that description, we know that
any place that Bob can swap into is also
a place that Bob can swap out of.

( We know that k+1 is also such a place.
( We know that,
( for each BEFORE and AFTER =< k+1,
( i ends BEFORE, j begins AFTER, j = i+1

Therefore, we know that
there is no place Bob is after all swaps.
There is no Bob-conservation for all swaps.

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
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 by: sergio - Fri, 13 May 2022 20:37 UTC

On 5/13/2022 3:27 PM, WM wrote:
> horand....@gmail.com schrieb am Freitag, 13. Mai 2022 um 19:18:45 UTC+2:
>> On Friday, 13 May 2022 at 14:01:08 UTC-3, WM wrote:
>>> William schrieb am Donnerstag, 12. Mai 2022 um 21:23:05 UTC+2:
>>>> On Thursday, May 12, 2022 at 10:07:27 AM UTC-3, WM wrote:
>>>> .>..[any single] element having an infinite set S(n) of successors can be removed.
>>>>
>>>> Indeed. Any single *element* of N_p can be removed.
>>> Together with all its predecessors!
>>>> (indeed any finite set of elements of N_p).
>>> There are no other sets defined by natural numbers. Every FISON is finite. No definable natural number is outside of a FISON.
>> *EVERY* natural number is an element of a FISON,
>
> No.

Liar. or provide a proof that natural number k is not in any FISONs

>
>> and *EVERY* natural number defines an end segment of cardinality aleph_0.
>
> No.

Liar. or provide a proof that natural number k is not in Endsegment(k).

>
> 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 - Fri, 13 May 2022 20:39 UTC

Jim Burns schrieb am Donnerstag, 12. Mai 2022 um 19:05:41 UTC+2:

> Do all the exchanges. No Bob.
> XXXXXX...

Wrong. You like to support magic. Fact is that you are in the position of a man who has collected some elements and claims to collect all the rest, but every time when he collects one element, he drops another element. So he will never collect all.
> In order for Bob to be somewhere after
> all the exchanges, there would need to be
> some place which Bob can exchange into and
> not out of.

If he got lost, then there must be a last place.
>
> Dark places, existing or not, have no effect
> because they are not places Bob can exchange into.
>
> Describe a place which Bob can exchange into.[1]
> From that description, we will know that
> it is also a place Bob can exchange out of.

That measn he will never settle. Maybe so, Then there is no "all" and no actual infinity.
>
> Therefore,
> there is no place which Bob can exchange into
> and not out of.
>
> Therefore,
> Bob is _not_ somewhere after all the exchanges.
> No Bob-conservation.

Then there is no "all exchanges".

> Dark places, existing or not, have no effect
> because they are not places Bob can exchange into.

After all transpositions (if all exist) there are as many places occupied as at the beginning. That is fact. But only those covered by X's are definable.

Regards, WM

Re: Natural numbers and vases

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From: inva...@invalid.com (sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Fri, 13 May 2022 15:39:29 -0500
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 by: sergio - Fri, 13 May 2022 20:39 UTC

On 5/13/2022 3:25 PM, WM wrote:
> FromTheRafters schrieb am Freitag, 13. Mai 2022 um 21:29:14 UTC+2:
>> WM used his keyboard to write :
>>> William schrieb am Donnerstag, 12. Mai 2022 um 21:23:05 UTC+2:
>>>> On Thursday, May 12, 2022 at 10:07:27 AM UTC-3, WM wrote:
>>>> .>..[any single] element having an infinite set S(n) of successors can be
>>>> removed.
>>>>
>>>> Indeed. Any single *element* of N_p can be removed.
>>>
>>> Together with all its predecessors!
>> That would be a proper subset, not a single element. You could use the
>> initial element to satisfy this trivially since it has no predecessor.
>>
>> Again, you are not removing them from a set
>
> I am removing elements from a set. A \ B is possible. Cantor does it too: wir könnten sie alsdann von S entfernen, ohne daß diese Menge dadurch die in 2) ausgedrückte Beziehung zu M verliert.
>
> Regards, WM

you snipped Cantor.

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 - Fri, 13 May 2022 20:41 UTC

Jim Burns schrieb am Donnerstag, 12. Mai 2022 um 19:35:52 UTC+2:

> > Describe a place which Bob can exchange into.[1]
> > From that description, we will know that
> > it is also a place Bob can exchange out of.

There is no fixed border between definable nd dark.
There is no end to potential infinity.

Regards, WM

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