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=107419&group=sci.math#107419

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From: inva...@invalid.com (Sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Tue, 26 Jul 2022 13:33:21 -0500
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by: Sergio - Tue, 26 Jul 2022 18:33 UTC

On 7/26/2022 10:35 AM, WM wrote:
> Sergio schrieb am Dienstag, 26. Juli 2022 um 14:13:38 UTC+2:
>> On 7/26/2022 6:10 AM, WM wrote:
>>> Sergio schrieb am Montag, 25. Juli 2022 um 21:14:36 UTC+2:
>>>> On 7/25/2022 12:09 PM, WM wrote:
>>>
>>>>> If every endsegment contains a number,
>>>> each endsegment contains an infinity of numbers.
>>>>> then, by inclusion monotony, all endsegments contain the same number.
>>>>> This is unavoidable.
>>>>
>>>> no, it is not possible.
>>>
>>> Take a leaking bathtub. As long as it contains a drop, this drop is the same for all states.
>>> As long as it contains many drops, at least one drop is the same for all states.
>>>
>> Bathtubs are finite, and do not apply to infinite sets.
>
> In mathematics drops can halve their size every time.

LAMO, you bought the wrong drops, you are trying to use your Bathtub Math to conjure up your shufflin darkies

>
> Regards, WM

Re: Natural numbers and vases III

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From: arm...@nifjianc.ii (Jim Africani)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Tue, 26 Jul 2022 19:09:16 -0000 (UTC)
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by: Jim Africani - Tue, 26 Jul 2022 19:09 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, 26 Jul 2022 19:28 UTC

On Tuesday, July 26, 2022 at 8:31:11 PM UTC+2, Sergio wrote:
> On 7/26/2022 10:40 AM, WM wrote:
> > Fritz Feldhase schrieb am Dienstag, 26. Juli 2022 um 14:26:53 UTC+2:
> >> On Tuesday, July 26, 2022 at 1:06:49 PM UTC+2, WM wrote:
> > > >
> > > > dark numbers are not subject to induction.
> > > >
> > > This implies that dark numbers aren't natural numbers, since all _natural numbers_ "are subject to induction".
> > >
> > > Proof (by induction). 1 is subject to induction. If a natural number n is subject to induction then n+1 is subject to induction too.
> > >
> > Of course.

Indeed!

> Wrong => if n is subject to post-induction than n+1 is is subject to post-induction too [...]

So what? But 1 is not subject to post-induction!

> > > Hence (by induction) all natural numbers are subject to induction. qed

Seems that Mückenheim cannot even handle proofs by induction any more, since he's willing to accepts P(1) and An e IN: P(n) -> P(n+1), but "questions" the conclusion An e IN: P(n). Holy shit!

Re: Natural numbers and vases III

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From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Tue, 26 Jul 2022 20:14 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, 27 Jul 2022 12:24 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, 27 Jul 2022 12:27 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, 27 Jul 2022 12:32 UTC

Fritz Feldhase schrieb am Dienstag, 26. Juli 2022 um 18:44:44 UTC+2:
> On Tuesday, July 26, 2022 at 5:45:54 PM UTC+2, WM wrote:
> > Gus Gassmann schrieb am Dienstag, 26. Juli 2022 um 13:16:42 UTC+2:
> > > On Tuesday, 26 July 2022 at 08:01:53 UTC-3, WM wrote:
> > > >
> > > > A simple example is the sequence 1 + 1/n.
> Better use a sequence of sets, say the sequence of intervalls (I_n) with I_n = (1, 1 + 1/n] for all n e IN.

I prefer to use the intervals J_n = [0, 1 + 1/n]. It models the sequence of infinite endsegment.
>
> This sequence is inclusion monotonic (since I_(n+1) c I_n for all n e IN).The Intersection of the terms of this sequence is empty. INTERSECTION_(n e IN) I_n = { }.

Of course. The intersection of my sequence however is [0, 1].

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 - Wed, 27 Jul 2022 12:39 UTC

Re: Natural numbers and vases III

<tbrdbi$k1p$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, 27 Jul 2022 08:08:01 -0500
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by: Sergio - Wed, 27 Jul 2022 13:08 UTC

On 7/27/2022 7:24 AM, WM wrote:
> Fritz Feldhase schrieb am Dienstag, 26. Juli 2022 um 18:32:30 UTC+2:
>
>> Proof (by induction). 1 is a natural number (by a Peano Axiom). If n is a natural number then n+1 is a natural number too (by a Peano Axiom and n + 1 = s(n)). Hence (by induction, another Peano Axiom) all natural numbers are natural numbers. qed
>>
>> Ok? :-)
>
> That i not provable by induction. Every definable natural number is subject to induction. But between each one and omega infinitely many remain. Nevertheless you cannot define a natural number which would remain, can you? Therefore
>
> Regards, WM
>

you see? Your "definable" numbers do not work in real Math.

In your definition of "definable" or "instantated" you equate beeps, honks, raps, colors of the rainbow and FISONs.

That foundation of your math is rotted to the core.

Re: Natural numbers and vases III

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From: inva...@invalid.com (Sergio)
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Subject: Re: Natural numbers and vases III
Date: Wed, 27 Jul 2022 08:10:08 -0500
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by: Sergio - Wed, 27 Jul 2022 13:10 UTC

On 7/27/2022 7:27 AM, WM wrote:
> William schrieb am Dienstag, 26. Juli 2022 um 18:39:12 UTC+2:
>> On Tuesday, July 26, 2022 at 12:41:46 PM UTC-3, WM wrote:
>>> William schrieb am Dienstag, 26. Juli 2022 um 15:45:38 UTC+2:
>>>
>>>> A stepwise process will not reach a final state.
>>> Nevertheless you cannot define a natural number which would remain, can you?.
>> So What? This does not mean the stepwise process reaches a final state.
>>
> Exactly. The stepwise process never reaches a final state. Infinitely many natural numbers prevent it, sitting between every defined n and omega. But you cannot define most of them. They are dark.
>
> Regards, WM

"dark" is simply a term you use for "unknown"

numbers do not "prevent" things at all.

Re: Natural numbers and vases III

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by: Sergio - Wed, 27 Jul 2022 13:11 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 - Wed, 27 Jul 2022 13:24 UTC

On Wednesday, 27 July 2022 at 09:39:31 UTC-3, WM wrote:
> Gus Gassmann schrieb am Dienstag, 26. Juli 2022 um 18:12:52 UTC+2:
> > On Tuesday, 26 July 2022 at 12:45:54 UTC-3, WM wrote:
> > [...]
> > > If you can't understand the abstract meaning, try to visualize it with geometric lengths.
> > If you can't express yourself in an understandable manner,
> I can. Students graduated from high school usually understand what I say. Perhaps you have to have to make up some lessons.

Nobody, least of all a stubborn, demented idiot like you, should judge their own language proficiency. That goes double for technical writing. Your attitude towards your own language skills does of course dovetail nicely with your repeated "proofs by intimidation", to which you regularly subject your poor students.

Re: Natural numbers and vases III

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by: FromTheRafters - Wed, 27 Jul 2022 13:42 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, 27 Jul 2022 14:09 UTC

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, 28 Jul 2022 05:26 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, 28 Jul 2022 14:21 UTC

Re: Natural numbers and vases III

<296d118d-8a74-4011-a81c-95bef56400fdn@googlegroups.com>

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Subject: Re: Natural numbers and vases III
From: horand.g...@gmail.com (Gus Gassmann)
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by: Gus Gassmann - Thu, 28 Jul 2022 14:34 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, 28 Jul 2022 16:30 UTC

Gus Gassmann schrieb am Donnerstag, 28. Juli 2022 um 16:34:27 UTC+2:
> On Thursday, 28 July 2022 at 11:21:44 UTC-3, WM wrote:
> > William schrieb am Mittwoch, 27. Juli 2022 um 16:09:17 UTC+2:
> > > On Wednesday, July 27, 2022 at 9:27:31 AM UTC-3, WM wrote:
> > > >... The stepwise process never reaches a final state.
> > >
> > > And, as you note, a final state exists.
> > >
> > Yes, if Cantor is right, but it can only be reached by collective action. Proof: By stepwise action it cannot be reached.
> Whatever "collective action" means to you

Collective action means to use "the set |N" or {1,2, 3, ...}. Individual action means to use 17 or 423.

> WHY do you act so surprised at that?

I am not surprise at that but about the claims that all natural numbers could be used indivdually.

Regards, WM

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 - Thu, 28 Jul 2022 17:12 UTC

Re: Natural numbers and vases III

<tbuioh$1ft9$1@gioia.aioe.org>

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From: inva...@invalid.com (Sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Thu, 28 Jul 2022 12:58:40 -0500
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by: Sergio - Thu, 28 Jul 2022 17:58 UTC

On 7/28/2022 9:21 AM, WM wrote:
> William schrieb am Mittwoch, 27. Juli 2022 um 16:09:17 UTC+2:
>> On Wednesday, July 27, 2022 at 9:27:31 AM UTC-3, WM wrote:
>>> ... The stepwise process never reaches a final state.
>>
>> And, as you note, a final state exists.
>>
> Yes, if Cantor is right,

Cantor IS right.

>but it can only be reached by collective action.

your "can only be reached by collective action" is meaningless in Math.

> Proof: By stepwise action it cannot be reached.

that is Spoof. "stepwise action" has no meaning in Math

>
> Regards, WM

use equations, it is the language of Math,

Re: Natural numbers and vases III

<tbuitv$1ft9$2@gioia.aioe.org>

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From: inva...@invalid.com (Sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Thu, 28 Jul 2022 13:01:34 -0500
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by: Sergio - Thu, 28 Jul 2022 18:01 UTC

On 7/28/2022 11:30 AM, WM wrote:
> Gus Gassmann schrieb am Donnerstag, 28. Juli 2022 um 16:34:27 UTC+2:
>> On Thursday, 28 July 2022 at 11:21:44 UTC-3, WM wrote:
>>> William schrieb am Mittwoch, 27. Juli 2022 um 16:09:17 UTC+2:
>>>> On Wednesday, July 27, 2022 at 9:27:31 AM UTC-3, WM wrote:
>>>>> ... The stepwise process never reaches a final state.
>>>>
>>>> And, as you note, a final state exists.
>>>>
>>> Yes, if Cantor is right, but it can only be reached by collective action. Proof: By stepwise action it cannot be reached.
>> Whatever "collective action" means to you
>
> Collective action means to use "the set |N" or {1,2, 3, ...}.

"Collective action" does not mean "set", one implies operation, the other implies an object

> Individual action means to use 17 or 423.

"Individual action " does not mean "17 or 423", one implies operation, the other implies an object

>
>> WHY do you act so surprised at that?
>
> I am not surprise at that but about the claims that all natural numbers could be used indivdually.

no one wants a Used Ant. Do you want a Used Ant ? Would you, Could you ?

>
> 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, 28 Jul 2022 20:31 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 - Thu, 28 Jul 2022 21:33 UTC

On Thursday, July 28, 2022 at 10:31:29 PM UTC+2, WM wrote:

> Now you have only to accept that every definable number can be defined (= reached by a stepwise process).

EACH AND EVERY natural number can be "reached by a stepwise process", you silly asshole full of shit.

Hint: If n is natural number then, starting with 0, n applications of the successor operation (aka steps) suffice.

Here's a simple example (if you can count to three - which is rather doubtful):

Take n = 3. Now:

0 => s0 => ss0 => sss0 .

Hint: s0 = 1, ss0 = s1 = 2, sss0 = ss1 = s2 = 3.

Hope this helps.

---------------------------------

Why are you so bad at math, Mückenheim?

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 - Thu, 28 Jul 2022 21:38 UTC

Don't panic.

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