novaBBS - sci.math - Re: Natural numbers and vases

Re: Natural numbers and vases

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https://www.novabbs.com/tech/article-flat.php?id=101527&group=sci.math#101527

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From: inva...@invalid.com (sergi o)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Mon, 30 May 2022 09:31:29 -0500
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by: sergi o - Mon, 30 May 2022 14:31 UTC

On 5/29/2022 10:56 AM, WM wrote:
> Jim Burns schrieb am Sonntag, 29. Mai 2022 um 01:11:12 UTC+2:
>> On 5/28/2022 3:52 PM, WM wrote:
>>> Jim Burns schrieb
>>> am Samstag, 28. Mai 2022 um 01:14:08 UTC+2:
>>>> This is possible because
>>>> there is no finite distance in which
>>>> every exchange of X and O happens.
>>>
>>> And there is no infinite distance and no completeness.
>> One kind of completeness is no last swap.
>
> That is not completeness. "Allow me to remark that the reality and the absolute principles of the integers appear to be much stronger than those of the world of sensations. And this fact has precisely one very simple reason, namely that the integers separately as well as in their actually infinite totality exist as eternal ideas in intellectu Divino in the highest degree of reality." [G. Cantor, letter to C. Hermite (30 Nov 1895)] Das God fail to think of the last part before omegain highest degree of reality?

quote has nothing to do with completness

>
>> Another kind is each swap described.
>
> And all swaps.
>>
>> We have the completeness of each swap described.
>> We have a description.
>
> Exchanging an X and an O from X.....O to O.....X will not make any of them disappear and will not make any of them multiply. It will simply maintain an X and an O.

therefore swaps do nothing.

I would never have believed before that matheologians are dishonest enough (because not even an earthworm can be stupid enough) to contradict this
simple fact.
>

you are the one that denies it, WM the pretend matheologian

> Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
Date: Mon, 30 May 2022 09:34:22 -0500
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by: sergi o - Mon, 30 May 2022 14:34 UTC

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 - Mon, 30 May 2022 15:58 UTC

horand....@gmail.com schrieb am Montag, 30. Mai 2022 um 16:32:32 UTC+2:
> On Sunday, 29 May 2022 at 17:36:47 UTC-3, WM wrote:
> > William schrieb am Sonntag, 29. Mai 2022 um 21:56:41 UTC+2:
> > > On Sunday, May 29, 2022 at 12:58:14 PM UTC-3, WM wrote:
> >
> > > >Exchanging an X and an O from X.....O to O.....X will not make any of them disappear and will not make any of them multiply. It will simply maintain
> > > two sets of cardinality aleph_0.
> > So it is! But: Cantor's claim is that the O's all will disappear.
> Bullshit! Cantor's "Zuordnungsprozess" (association process) is not a *stepwise* process.

Counting, abzählen, is a stepwise process. At least every critic is allowed to check every number. The attempt to deny this shows the dishonesty of matheologians.

> The entire mapping exists as a whole.

Then there would exist a last one. Complete existence without proving completeness is the basic fault of matheology.

> Every pair (n,m) is associated with a unique k by Cantor's formula.

But every definable k has aleph_0 sucessors, aleph_0 of which are undefinable. Because every definable k has aleph_0 sucessors, aleph_0 of which are undefinable. Because every definable k has aleph_0 sucessors, aleph_0 of which are undefinable.

Regards, WM

Re: Natural numbers and vases

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by: WM - Mon, 30 May 2022 16:45 UTC

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
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by: WM - Mon, 30 May 2022 16:45 UTC

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
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by: WM - Mon, 30 May 2022 16:50 UTC

Re: Natural numbers and vases

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From: nom...@afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Mon, 30 May 2022 14:10:48 -0400
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by: FromTheRafters - Mon, 30 May 2022 18:10 UTC

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: Mon, 30 May 2022 14:12:03 -0400
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by: FromTheRafters - Mon, 30 May 2022 18:12 UTC

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 - Mon, 30 May 2022 18:23 UTC

Re: Natural numbers and vases

<b8469877-db01-4387-23c7-99e45eaea0e4@att.net>

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https://www.novabbs.com/tech/article-flat.php?id=101564&group=sci.math#101564

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Mon, 30 May 2022 14:54:45 -0400
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by: Jim Burns - Mon, 30 May 2022 18:54 UTC

On 5/30/2022 12:50 PM, WM wrote:
> Jim Burns schrieb
> am Sonntag, 29. Mai 2022 um 23:33:51 UTC+2:

>> This simple fact?
>>
>> 1/1:1/1 1/2:2/1 1/3:4/1 1/4:7/1 ...
>> 2/1:3/1 2/2:5/1 2/3:8/1 2/4:12/1 ...
>> 3/1:6/1 3/2:9/1 3/3:13/1 3/4:18/1 ...
>> 4/1:10/1 4/2:14/1 4/3:19/1 4/4:25/1 ...
>> 5/1:15/1 5/2:20/1 5/3:26/1 5/4:33/1 ...
>> 6/1:21/1 6/2:27/1 6/3:34/1 6/4:42/1 ...
>> 7/1:28/1 7/2:35/1 7/3:43/1 7/4:52/1 ...
>> 8/1:36/1 8/2:44/1 8/3:53/1 8/4:63/1 ...
>> 9/1:45/1 9/2:54/1 9/3:64/1 9/4:75/1 ...
>> ... ... ... ...
>
> You need the ... in order to veil the fact that
> all your indices have aleph_0 successors.

I don't _need_ the ... at all.
I provided them as a courtesy to you.
It looks like you strongly favor
writing things out like that.

Without the ...

The matrix is the collection of m/n:k/1
such that k = (m+n-1)*(m+n-2)/2+m
and,
for each BEFORE and AFTER =< m
and each BEFORE and AFTER =< n,
some j ends BEFORE and j+1 begins AFTER.

For each m and n _like that_
we know that k is also _like that_
Thus
k/1 is in the first column of the matrix.

I can unveil more facts, and more, and more,
limited only by my patience and the patience
of my hypothetical reader, to the point that
even a gaggle of transistors, such as found
in a computer, would understand and agree.

> But my proof "Exchanging an X and an O
> from X.....O to O.....X will not make any of them
> disappear and will not make any of them multiply"
> is independent of successors but holds in infinity.

If the swaps do not stop, no O remains.

....because
each place in the matrix which an O can get to
is a place it can get away from.

Re: Natural numbers and vases

<45eae0c2-87d5-c011-ed8b-63041fbe2e24@att.net>

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Mon, 30 May 2022 15:01:02 -0400
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by: Jim Burns - Mon, 30 May 2022 19:01 UTC

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 - Tue, 31 May 2022 11:59 UTC

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 - Tue, 31 May 2022 12:50 UTC

FromTheRafters schrieb am Montag, 30. Mai 2022 um 20:11:08 UTC+2:
> WM has brought this to us :
> > FromTheRafters schrieb am Montag, 30. Mai 2022 um 00:08:12 UTC+2:
> >> WM expressed precisely :
> >>> William schrieb am Sonntag, 29. Mai 2022 um 21:56:41 UTC+2:
> >>>> On Sunday, May 29, 2022 at 12:58:14 PM UTC-3, WM wrote:
> >>>>> Exchanging an X and an O from X.....O to O.....X will not make any of
> >>>>> them disappear and will not make any of them multiply. It will simply
> >>>>> maintain
> >>>> two sets of cardinality aleph_0.
> >>>
> >>> So it is! But: Cantor's claim is that the O's all will disappear.
> >> Where has he claimed this?
> >
> > In many places, for instance here: "Wenn zwei wohldefinierte
> > Mannigfaltigkeiten M und N sich eindeutig und vollständig, Element für
> > Element, einander zuordnen lassen (was, wenn es auf eine Art möglich ist,
> > immer auch noch auf viele andere Weisen geschehen kann), so möge für das
> > Folgende die Ausdrucksweise gestattet sein, daß diese Mannigfaltigkeiten
> > gleiche Mächtigkeit haben, oder auch, daß sie äquivalent sind." [Cantor, p.
> > 119]
> What is German for Xs and Os?

X is my abbreviation for IndeX, and O stand for Ohne Index.
Indexing is same as counting or abzählen or zuordnen where one set is |N.

Cantor claims zuordnen, that means removing the O's by X's.

Regards, WM

Re: Natural numbers and vases

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by: WM - Tue, 31 May 2022 12:52 UTC

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
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Re: Natural numbers and vases

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Date: Tue, 31 May 2022 06:05:51 -0700 (PDT)
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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Tue, 31 May 2022 13:05 UTC

Re: Natural numbers and vases

<t7552d$11eu$2@gioia.aioe.org>

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From: inva...@invalid.com (sergi o)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Tue, 31 May 2022 08:26:37 -0500
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by: sergi o - Tue, 31 May 2022 13:26 UTC

Re: Natural numbers and vases

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https://www.novabbs.com/tech/article-flat.php?id=101626&group=sci.math#101626

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Subject: Re: Natural numbers and vases
Date: Tue, 31 May 2022 08:28:45 -0500
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by: sergi o - Tue, 31 May 2022 13:28 UTC

Re: Natural numbers and vases

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From: inva...@invalid.com (sergi o)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Tue, 31 May 2022 08:30:29 -0500
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by: sergi o - Tue, 31 May 2022 13:30 UTC

On 5/31/2022 7:50 AM, WM wrote:
> FromTheRafters schrieb am Montag, 30. Mai 2022 um 20:11:08 UTC+2:
>> WM has brought this to us :
>>> FromTheRafters schrieb am Montag, 30. Mai 2022 um 00:08:12 UTC+2:
>>>> WM expressed precisely :
>>>>> William schrieb am Sonntag, 29. Mai 2022 um 21:56:41 UTC+2:
>>>>>> On Sunday, May 29, 2022 at 12:58:14 PM UTC-3, WM wrote:
>>>>>>> Exchanging an X and an O from X.....O to O.....X will not make any of
>>>>>>> them disappear and will not make any of them multiply. It will simply
>>>>>>> maintain
>>>>>> two sets of cardinality aleph_0.
>>>>>
>>>>> So it is! But: Cantor's claim is that the O's all will disappear.
>>>> Where has he claimed this?
>>>
>>> In many places, for instance here: "Wenn zwei wohldefinierte
>>> Mannigfaltigkeiten M und N sich eindeutig und vollständig, Element für
>>> Element, einander zuordnen lassen (was, wenn es auf eine Art möglich ist,
>>> immer auch noch auf viele andere Weisen geschehen kann), so möge für das
>>> Folgende die Ausdrucksweise gestattet sein, daß diese Mannigfaltigkeiten
>>> gleiche Mächtigkeit haben, oder auch, daß sie äquivalent sind." [Cantor, p.
>>> 119]
>> What is German for Xs and Os?
>
> X is my abbreviation for IndeX, and O stand for Ohne Index.
> Indexing is same as counting

no, it is not.

>
> Cantor claims zuordnen, that means removing the O's by X's.

where does Cantor say X and O's ? or are you just making stuff up ?

>
> Regards, WM

Re: Natural numbers and vases

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From: nom...@afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Tue, 31 May 2022 10:04:56 -0400
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by: FromTheRafters - Tue, 31 May 2022 14:04 UTC

Re: Natural numbers and vases

<t757tl$9e7$1@dont-email.me>

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From: nom...@afraid.org (FromTheRafters)
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Subject: Re: Natural numbers and vases
Date: Tue, 31 May 2022 10:16:21 -0400
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by: FromTheRafters - Tue, 31 May 2022 14:16 UTC

Re: Natural numbers and vases

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From: james.g....@att.net (Jim Burns)
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Subject: Re: Natural numbers and vases
Date: Tue, 31 May 2022 10:23:27 -0400
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by: Jim Burns - Tue, 31 May 2022 14:23 UTC

Re: Natural numbers and vases

<t7594g$19i2$1@gioia.aioe.org>

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Subject: Re: Natural numbers and vases
Date: Tue, 31 May 2022 09:35:58 -0500
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by: sergi o - Tue, 31 May 2022 14:35 UTC

On 5/31/2022 9:04 AM, FromTheRafters wrote:
> WM explained :
>> FromTheRafters schrieb am Montag, 30. Mai 2022 um 20:09:53 UTC+2:
>>> WM explained on 5/30/2022 :
>>>> Jim Burns schrieb am Sonntag, 29. Mai 2022 um 23:33:51 UTC+2:
>>>>>
>>>>> This simple fact?
>>>>> 1/1:1/1 1/2:2/1 1/3:4/1 1/4:7/1 ... 2/1:3/1 2/2:5/1 2/3:8/1 2/4:12/1 ... 3/1:6/1 3/2:9/1 3/3:13/1 3/4:18/1 ... 4/1:10/1 4/2:14/1 4/3:19/1 4/4:25/1
>>>>> ... 5/1:15/1 5/2:20/1 5/3:26/1 5/4:33/1 ... 6/1:21/1 6/2:27/1 6/3:34/1 6/4:42/1 ... 7/1:28/1 7/2:35/1 7/3:43/1 7/4:52/1 ... 8/1:36/1 8/2:44/1
>>>>> 8/3:53/1 8/4:63/1 ... 9/1:45/1 9/2:54/1 9/3:64/1 9/4:75/1 ... ... ... ... ...
>>>>
>>>> You need the ... in order to veil the fact that all your indices have aleph_0 successors.
>>> Veil? It rather points out the fact.
>>
>> It feigns that all were definable.
>
> The fact that they are defined implies that they were in fact definable.

WM needs to further define define;

defining (in process of)

re-defined, re-defining (re consideration of)

post-defined, post definable

pre-defined, pre definable

undefined, undefining

not defining, not definable

reversion of a defined number to undefined state

lost defined status

stickers to put onto defined numbers saying they are defined

Re: Natural numbers and vases

<t75999$m95$1@dont-email.me>

copy mid

https://www.novabbs.com/tech/article-flat.php?id=101634&group=sci.math#101634

copy link Newsgroups: sci.math

Path: i2pn2.org!i2pn.org!eternal-september.org!reader02.eternal-september.org!.POSTED!not-for-mail
From: nom...@afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Tue, 31 May 2022 10:39:37 -0400
Organization: Peripheral Visions
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by: FromTheRafters - Tue, 31 May 2022 14:39 UTC

sergi o wrote :
> On 5/31/2022 7:50 AM, WM wrote:
>> FromTheRafters schrieb am Montag, 30. Mai 2022 um 20:11:08 UTC+2:
>>> WM has brought this to us :
>>>> FromTheRafters schrieb am Montag, 30. Mai 2022 um 00:08:12 UTC+2:
>>>>> WM expressed precisely :
>>>>>> William schrieb am Sonntag, 29. Mai 2022 um 21:56:41 UTC+2:
>>>>>>> On Sunday, May 29, 2022 at 12:58:14 PM UTC-3, WM wrote:
>>>>>>>> Exchanging an X and an O from X.....O to O.....X will not make any of
>>>>>>>> them disappear and will not make any of them multiply. It will simply
>>>>>>>> maintain
>>>>>>> two sets of cardinality aleph_0.
>>>>>>
>>>>>> So it is! But: Cantor's claim is that the O's all will disappear.
>>>>> Where has he claimed this?
>>>>
>>>> In many places, for instance here: "Wenn zwei wohldefinierte
>>>> Mannigfaltigkeiten M und N sich eindeutig und vollständig, Element für
>>>> Element, einander zuordnen lassen (was, wenn es auf eine Art möglich ist,
>>>> immer auch noch auf viele andere Weisen geschehen kann), so möge für das
>>>> Folgende die Ausdrucksweise gestattet sein, daß diese Mannigfaltigkeiten
>>>> gleiche Mächtigkeit haben, oder auch, daß sie äquivalent sind." [Cantor,
>>>> p.
>>>> 119]
>>> What is German for Xs and Os?
>>
>> X is my abbreviation for IndeX, and O stand for Ohne Index.
>> Indexing is same as counting
>
> no, it is not.
>
>
>
>>
>> Cantor claims zuordnen, that means removing the O's by X's.
>
> where does Cantor say X and O's ? or are you just making stuff up ?
>
Those are apparently his new O "dark indices" which don't exist until
you X kiss them. Quite an elaborate hallucination assuming that they
cannot be 'unkissed' creates. So many virgins, so little time, and
always only a finite number of kissed X cells and NaN (to WM apparently
just a very large natural number) of Os to go.

It's kind of like rearranging table scraps from a Scrooge McDuck
fricassée into a sequence of matrices and thinking it means something.

"He don't know me vewy well, DO he?" -- Bugs Bunny

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

Subject	Author
Natural numbers and vases	William
Re: Natural numbers and vases	Ross A. Finlayson
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	FromTheRafters
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Jim Burns
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Jim Burns
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Jim Burns
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Jim Burns
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Chet Hirasi
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	Chet Hirasi
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	FredJeffries
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	FromTheRafters
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Jim Burns
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Ross A. Finlayson
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	Mostowski Collapse
Re: Natural numbers and vases	Socratis T.n.p.