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

Re: Natural numbers and vases III

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https://www.novabbs.com/tech/article-flat.php?id=108639&group=sci.math#108639

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From: inva...@invalid.com (Sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Tue, 9 Aug 2022 08:36:59 -0500
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by: Sergio - Tue, 9 Aug 2022 13:36 UTC

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Tue, 9 Aug 2022 13:52 UTC

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: horand.g...@gmail.com (Gus Gassmann)
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by: Gus Gassmann - Tue, 9 Aug 2022 14:04 UTC

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: wpihug...@gmail.com (William)
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by: William - Tue, 9 Aug 2022 14:33 UTC

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: zelos.ma...@gmail.com (zelos...@gmail.com)
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by: zelos...@gmail.com - Wed, 10 Aug 2022 05:52 UTC

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Wed, 10 Aug 2022 15:32 UTC

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Wed, 10 Aug 2022 15:36 UTC

Re: Natural numbers and vases III

<5cbcf710-1375-4d52-9418-c3d6493d7b9bn@googlegroups.com>

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https://www.novabbs.com/tech/article-flat.php?id=108733&group=sci.math#108733

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Wed, 10 Aug 2022 15:41 UTC

Re: Natural numbers and vases III

<d3e204fd-bd43-4e5b-b296-357e133f637en@googlegroups.com>

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https://www.novabbs.com/tech/article-flat.php?id=108740&group=sci.math#108740

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Subject: Re: Natural numbers and vases III
From: horand.g...@gmail.com (Gus Gassmann)
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by: Gus Gassmann - Wed, 10 Aug 2022 16:56 UTC

Re: Natural numbers and vases III

<b903ab58-92cb-4107-8437-297baf990898n@googlegroups.com>

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https://www.novabbs.com/tech/article-flat.php?id=108741&group=sci.math#108741

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Subject: Re: Natural numbers and vases III
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Wed, 10 Aug 2022 16:56 UTC

Re: Natural numbers and vases III

<3f7a7385-0b7c-48d5-97c3-e3c41251f2c3n@googlegroups.com>

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https://www.novabbs.com/tech/article-flat.php?id=108743&group=sci.math#108743

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Subject: Re: Natural numbers and vases III
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Wed, 10 Aug 2022 16:59 UTC

Re: Natural numbers and vases III

<fcd89499-05a7-4477-9d63-d3e12ad78943n@googlegroups.com>

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https://www.novabbs.com/tech/article-flat.php?id=108746&group=sci.math#108746

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Subject: Re: Natural numbers and vases III
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Wed, 10 Aug 2022 17:06 UTC

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: wpihug...@gmail.com (William)
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by: William - Wed, 10 Aug 2022 17:32 UTC

Re: Natural numbers and vases III

<td0rjj$1u1$1@gioia.aioe.org>

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From: inva...@invalid.com (Sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Wed, 10 Aug 2022 12:58:08 -0500
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by: Sergio - Wed, 10 Aug 2022 17:58 UTC

On 8/10/2022 12:06 PM, Fritz Feldhase wrote:
> On Wednesday, August 10, 2022 at 6:56:22 PM UTC+2, Gus Gassmann wrote:
>> On Wednesday, 10 August 2022 at 12:32:16 UTC-3, WM wrote:
>>> Gus Gassmann schrieb am Dienstag, 9. August 2022 um 14:15:21 UTC+2:
>>>> On Tuesday, 9 August 2022 at 07:51:27 UTC-3, WM wrote:
>>>>>
>>>>> XOOOO...
>>>>> XOOOO...
>>>>> XOOOO...
>>>>> XOOOO...
>>>>> XOOOO...
>>>>> ...
>>>>>
>>>>> will never become
>>>>>
>>>>> XXX...
>>>>> XXX...
>>>>> XXX...
>>>>> ...
>>>>>
>>>>> by exchanges of X and O.
>>>>
>>>> Because you say so?
>>>>
>>> No, because never an O will disappear.
>>>
>> Hence, because you say so.
>
> No, because he is right.
>
> Hint: If there is an O in the matrix A then there's an O in the matrix swap(i,j;k,l)(A).
>
>> I talked about a *limit* [...]
>
> In contrast to WM.
>
> Starting with the "original" matrix NO swap will "result in" the limit matrix. Right?
>
> (You see, I don't think that we should talk about "supertasks" in this context.)

well, WM is outside of Math when he swapadoodles, but swaps avoid Math, and avoids making new sets and butchers the set of rationals in a meaningless
way. It is intentional misleading confusitorium by WM.

there is the set of natural numbers, and the set of rationals (which can be arranged into a matrix), and a one to one mapping between the two. That is
easy to verify and checkout. However, WMs matrix-cide swaparoofestivle disagrees with that, so *WM is wrong*. Where? its unimportant.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
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by: Gus Gassmann - Wed, 10 Aug 2022 18:46 UTC

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
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by: Gus Gassmann - Wed, 10 Aug 2022 18:55 UTC

On Wednesday, 10 August 2022 at 14:06:51 UTC-3, Fritz Feldhase wrote:
> On Wednesday, August 10, 2022 at 6:56:22 PM UTC+2, Gus Gassmann wrote:
> > On Wednesday, 10 August 2022 at 12:32:16 UTC-3, WM wrote:
> > > Gus Gassmann schrieb am Dienstag, 9. August 2022 um 14:15:21 UTC+2:
> > > > On Tuesday, 9 August 2022 at 07:51:27 UTC-3, WM wrote:
> > > > >
> > > > > XOOOO...
> > > > > XOOOO...
> > > > > XOOOO...
> > > > > XOOOO...
> > > > > XOOOO...
> > > > > ...
> > > > >
> > > > > will never become
> > > > >
> > > > > XXX...
> > > > > XXX...
> > > > > XXX...
> > > > > ...
> > > > >
> > > > > by exchanges of X and O.
> > > >
> > > > Because you say so?
> > > >
> > > No, because never an O will disappear.
> > >
> > Hence, because you say so.
> No, because he is right.
> Hint: If there is an O in the matrix A then there's an O in the matrix swap(i,j;k,l)(A).
> > I talked about a *limit* [...]
>
> In contrast to WM.
>
> Starting with the "original" matrix NO swap will "result in" the limit matrix. Right?
>
> (You see, I don't think that we should talk about "supertasks" in this context.)

Is the taking of a limit a "supertask"? I happen to think not, and I do not want to give anybody any cover to try to claim that taking the limit of 1/n as n-->oo is a supertask and hence not possible. Maybe I am a radical, but when I see a map from |N to A, I always try to visualize a limiting expression or object. With any of WM's processes that he claims prove the inconsistency of ZFC, the limit is self-evident. He does not have any idea how to show an inconsistency, anyway, and I do not want to encourage him in his lunacy.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Thu, 11 Aug 2022 00:02 UTC

On Wednesday, August 10, 2022 at 8:55:33 PM UTC+2, Gus Gassmann wrote:
> On Wednesday, 10 August 2022 at 14:06:51 UTC-3, Fritz Feldhase wrote:
> > On Wednesday, August 10, 2022 at 6:56:22 PM UTC+2, Gus Gassmann wrote:
> > > On Wednesday, 10 August 2022 at 12:32:16 UTC-3, WM wrote:
> > > > Gus Gassmann schrieb am Dienstag, 9. August 2022 um 14:15:21 UTC+2:
> > > > > On Tuesday, 9 August 2022 at 07:51:27 UTC-3, WM wrote:
> > > > > >
> > > > > > XOOOO...
> > > > > > XOOOO...
> > > > > > XOOOO...
> > > > > > XOOOO...
> > > > > > XOOOO...
> > > > > > ...
> > > > > >
> > > > > > will never become
> > > > > >
> > > > > > XXX...
> > > > > > XXX...
> > > > > > XXX...
> > > > > > ...
> > > > > >
> > > > > > by exchanges of X and O.
> > > > >
> > > > > Because you say so?
> > > > >
> > > > No, because never an O will disappear.
> > > >
> > > Hence, because you say so.
> > No, because he is right.
> > Hint: If there is an O in the matrix A then there's an O in the matrix swap(i,j;k,l)(A).
> > > I talked about a *limit* [...]
> >
> > In contrast to WM.
> >
> > Starting with the "original" matrix NO swap will "result in" the limit matrix. Right?
> >
> > (You see, I don't think that we should talk about "supertasks" in this context.)
> >
> Is the taking of a limit a "supertask"?

Of course not.

> I happen to think not, and I do not want to give anybody any cover to try to claim that taking the limit of 1/n as n-->oo is a supertask and hence not possible.

Well, its rather simple: It isn't. It's just a certain mathematical approach to "deal" with (certain) sequences and functions "in a certain way". Actually, (besides other things) it allows to "model" (handle, deal with) "physical phenomena" with mathematical means, especially movement (which itself might be "described" as a "supertask").

> Maybe I am a radical, but when I see a map from |N to A, I always try to visualize a limiting expression or object.

Ah. Never had that desire. Sure when considering "infinite series" it's rather inevitable (if the series converges).

1/2 + 1/4 + 1/8 + ...

may be considered the limit (...) in this case.

> With any of WM's processes that he claims prove the inconsistency of ZFC,

C'mon. You know that "this approach" is just nonsense.

> the limit is self-evident. He does not have any idea how to show an inconsistency, anyway, and I do not want to encourage him in his lunacy.

Of course.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Thu, 11 Aug 2022 00:17 UTC

On Wednesday, August 10, 2022 at 8:46:32 PM UTC+2, Gus Gassmann wrote:
> On Wednesday, 10 August 2022 at 14:06:51 UTC-3, Fritz Feldhase wrote:
> >
> > Starting with the "original" matrix NO swap will "result in" the limit matrix. Right?
> >
> In the sense that the matrix of all X is one of the matrices in the sequence of exchanges, of course not.

Right.

> But that does not, of course, invalidate the Cantor enumeration of the positive rationals, and I do not want to give the impression that I at any time support WM's idiotic contention to the contrary.

Sure. I get/got that.

But WM is still right when he's stressing the point that the matrix

> > > > > XXX...
> > > > > XXX...
> > > > > XXX...
> > > > > ...
> > > > >

cannot be "reached" by a stepwise process of "swaps" starting with the matrix

> > > > > XOOOO...
> > > > > XOOOO...
> > > > > XOOOO...
> > > > > XOOOO...
> > > > > XOOOO...
> > > > > ...

(at least not if we model that process by a sequence).

On the other hand, THERE SIMPLEY IS NO NEED FOR THAT. :-)

It's nice to see that we can actually get

> > > > > XXX...
> > > > > XXX...
> > > > > XXX...
> > > > > ...
> > > > >

as the limit of this sequence (decribed by Mückenheim).

But what do we GAIN by that? Cantor's "argument" does not need this fact. It does not refer to a "limit of a sequence" descibing the results of swaps)..

Cantor just defined a certain sequence (and proved certain properies of that sequence), that's all.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: horand.g...@gmail.com (Gus Gassmann)
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by: Gus Gassmann - Thu, 11 Aug 2022 02:31 UTC

Re: Natural numbers and vases III

<td1tfe$15ou$1@gioia.aioe.org>

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Subject: Re: Natural numbers and vases III
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by: Sergio - Thu, 11 Aug 2022 03:36 UTC

On 8/10/2022 9:31 PM, Gus Gassmann wrote:
> On Wednesday, 10 August 2022 at 21:17:38 UTC-3, Fritz Feldhase wrote:
> [...]
>> It's nice to see that we can actually get
>>
>>>>>>> XXX...
>>>>>>> XXX...
>>>>>>> XXX...
>>>>>>> ...
>>>>>>>
>>
>> as the limit of this sequence (decribed by Mückenheim).
>>
>> But what do we GAIN by that? Cantor's "argument" does not need this fact. It does not refer to a "limit of a sequence" descibing the results of swaps).
>
> I like to use it to stick another pin into WM's balloon. I did not invent the matrix sequence, but WM used it to convince himself that it disproves Cantor. I am trying to show him that you can get results from it that agree with Cantor.

If it is not in agreement with Cantor, it's wrong. Cantor's diagonal proof is so simple. It would have been found by someone else back then in a few
years. WMs matrix of rationals swaparoo fest is a muddy mess, and has errors in it, since it disagrees with a known good simple proof.
His math is like a tar baby, always a mess, and you get gooky if you touch it.

>
>> Cantor just defined a certain sequence (and proved certain properies of that sequence), that's all.
>
> Yes, and it is clear that WM can't wrap his tiny birdbrain around it. But he accepts limits, at least in some forms, so my hope was/is that by making him look at the limiting behaviour of other sequences he'd actually figure something out. JB, as far as I understand his motivation, is trying to do the same thing - in a different way. It probably is hopeless (vergebliche Liebesmüh', I guess).
>

WMs just trying to troll here. Cantor is name recognition to attract others.
It does seem that he can only deal with one thing/idea at a time though, and poorly.

He does not honor Math, he disparages it.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
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by: zelos...@gmail.com - Thu, 11 Aug 2022 05:10 UTC

Re: Natural numbers and vases III

<td2mi7$259l1$1@dont-email.me>

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From: FTR...@nomail.afraid.org (FromTheRafters)
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Subject: Re: Natural numbers and vases III
Date: Thu, 11 Aug 2022 06:44:17 -0400
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by: FromTheRafters - Thu, 11 Aug 2022 10:44 UTC

Gus Gassmann used his keyboard to write :
> On Wednesday, 10 August 2022 at 21:17:38 UTC-3, Fritz Feldhase wrote:
> [...]
>> It's nice to see that we can actually get
>>
>>>>>>> XXX...
>>>>>>> XXX...
>>>>>>> XXX...
>>>>>>> ...
>>>>>>>
>>
>> as the limit of this sequence (decribed by Mückenheim).
>>
>> But what do we GAIN by that? Cantor's "argument" does not need this fact. It
>> does not refer to a "limit of a sequence" descibing the results of swaps).
>
> I like to use it to stick another pin into WM's balloon. I did not invent the
> matrix sequence, but WM used it to convince himself that it disproves Cantor.
> I am trying to show him that you can get results from it that agree with
> Cantor.
>
>> Cantor just defined a certain sequence (and proved certain properies of that
>> sequence), that's all.
>
> Yes, and it is clear that WM can't wrap his tiny birdbrain around it. But he
> accepts limits, at least in some forms, so my hope was/is that by making him
> look at the limiting behaviour of other sequences he'd actually figure
> something out. JB, as far as I understand his motivation, is trying to do the
> same thing - in a different way. It probably is hopeless (vergebliche
> Liebesmüh', I guess). But it is better than just using the same three
> profanities all the time.

ISTM that he disproves his theory every time he takes on a new
discipline to 'explain' it. This time he seems to have an N by N table
and in an attempt to disprove Cantor ends up with more 'numbers' in the
infinitely many permutations of his table than there are natural
numbers.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
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by: WM - Thu, 11 Aug 2022 12:49 UTC

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Thu, 11 Aug 2022 12:50 UTC

Fritz Feldhase schrieb am Mittwoch, 10. August 2022 um 19:06:51 UTC+2:
> On Wednesday, August 10, 2022 at 6:56:22 PM UTC+2, Gus Gassmann wrote:
> > On Wednesday, 10 August 2022 at 12:32:16 UTC-3, WM wrote:
> > > Gus Gassmann schrieb am Dienstag, 9. August 2022 um 14:15:21 UTC+2:
> > > > On Tuesday, 9 August 2022 at 07:51:27 UTC-3, WM wrote:
> > > > >
> > > > > XOOOO...
> > > > > XOOOO...
> > > > > XOOOO...
> > > > > XOOOO...
> > > > > XOOOO...
> > > > > ...
> > > > >
> > > > > will never become
> > > > >
> > > > > XXX...
> > > > > XXX...
> > > > > XXX...
> > > > > ...
> > > > >
> > > > > by exchanges of X and O.
> > > >
> > > > Because you say so?
> > > >
> > > No, because never an O will disappear.
> > >
> > Hence, because you say so.
> No, because he is right.
> Hint: If there is an O in the matrix A then there's an O in the matrix swap(i,j;k,l)(A).
> > I talked about a *limit* [...]
>
> In contrast to WM.
>
> Starting with the "original" matrix NO swap will "result in" the limit matrix. Right?
>
> (You see, I don't think that we should talk about "supertasks" in this context.)

Cantor used it. He said that every step can be defined. "Wenn zwei wohldefinierte Mannigfaltigkeiten M und N sich eindeutig und vollständig, Element für Element, einander zuordnen lassen" [Cantor, p. 119]

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Thu, 11 Aug 2022 12:52 UTC

There are running jobs. Why don't you go chase them?

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

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

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