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

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, 3 Aug 2022 17:36 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 - Wed, 3 Aug 2022 18:49 UTC

On Wednesday, August 3, 2022 at 7:18:29 PM UTC+2, Gus Gassmann wrote:
> On Wednesday, 3 August 2022 at 14:10:56 UTC-3, Fritz Feldhase wrote:
> > On Wednesday, August 3, 2022 at 5:22:57 PM UTC+2, William wrote:
> > >
> > > Note that even if we accept [...] that ℕ_def changes, it [...]
> > >
> > Hint: "sets are fixed they do not change" (Sergio)
> >
> > Indeed!
> >
> > Hint: If a !e ℕ_def then ℕ_def u {a} =/= N_def.
> >
> > In the same way, if a e ℕ_def then ℕ_def \ {a} =/= N_def.
> >
> > Got that, dumbo?
> >
> > Sets don't GROW or SHRINK.
> >
> > Actually, sets are "specified" ("defined") by there elements. If the term "N_def" refers to a certain set IN A CERTAIN CONTEXT, then you should not adopt "IN_def" to refer to a different set IN THE SAME CONTEXT. This will lead to inconsistencies. (In the context of math/logic this holds for names/terms in general.)
> >
> > This means: If (in a certain context) you call a certain set which does not contain the element /a/ "N_def". You should not call a set which DOES contain /a/ as an element "N_def" too (in the same context).
> >
> > After all, this would allow to derive "~(a e N_def) & a e N_def" - a contradiction.
> >
> But isn't such an "inconsistency" exactly what he is looking for? (Please don't hit me...)

Indeed! AND NOW ... RUN!!!!!!!!

:-)

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, 3 Aug 2022 21:55 UTC

William schrieb am Mittwoch, 3. August 2022 um 19:36:41 UTC+2:
> On Wednesday, August 3, 2022 at 12:33:39 PM UTC-3, WM wrote:

> > > > They may have other properties of natural numbers. But we cannot know.
> > > Of course we can There are many ways, e.g. induction, to know that a number which does not have the property "can be written down", has other properties.
> > Many.
> And even if n cannot be written down you "can know" that n has each of these properties.

Not all, however. The property that a number can be written down (independent of time and ressources) is not a property of these numbers. There is a stronger obstacle than lack of time and ressources.

For instance you cannot name a number n the endsegment E(n) of which cannot removed from ∩{E(k) : k ∈ ℕ} = { } in order to maintain the empty result. Every endsegment which you can name can be removed. That means the endsegments which cause the empty intersection cannot be defined, not even in principle. Why should their other properties be the same?

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
Date: Wed, 3 Aug 2022 17:04:54 -0500
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by: Sergio - Wed, 3 Aug 2022 22:04 UTC

On 8/3/2022 4:55 PM, WM wrote:
> William schrieb am Mittwoch, 3. August 2022 um 19:36:41 UTC+2:
>> On Wednesday, August 3, 2022 at 12:33:39 PM UTC-3, WM wrote:
>
>>>>> They may have other properties of natural numbers. But we cannot know.
>>>> Of course we can There are many ways, e.g. induction, to know that a number which does not have the property "can be written down", has other properties.
>>> Many.
>> And even if n cannot be written down you "can know" that n has each of these properties.
>
> Not all, however. The property that a number can be written down (independent of time and ressources) is not a property of these numbers. There is a stronger obstacle than lack of time and ressources.

Answer: you dont know the number, it is unknown. Use a variable, like k for it. (all done)

>
> For instance you cannot name a number n the endsegment E(n) of which cannot removed from ∩{E(k) : k ∈ ℕ} = { } in order to maintain the empty result.

Rather, what is in the intersection of all but one endsegments, say E(1) is not in the intersection....

since E(1) is E(2) + {1}, is 1 in the intersection or not ? No.

therefore, I can name any n, it is not in the intersection, and your above statement is 100% poopus.

> Every endsegment which you can name can be removed. That means the endsegments which cause the empty intersection cannot be defined, not even in principle.

grasping at dark straws time...

> Why should their other properties be the same?

they are.

>
> 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 - Wed, 3 Aug 2022 22:41 UTC

On Wednesday, August 3, 2022 at 6:56:00 PM UTC-3, WM wrote:
> The property that a number can be written down (independent of time and ressources) is not a property of these numbers.

Piffle.

1 can be written down.
if n can be written down then n+1 can be written down (independent of time and resources)

Hence each element of |N (without exception) can be written down (independent of time and resources).

> .. For instance you cannot name a number n the endsegment E(n) of which cannot removed from ∩{E(k) : k ∈ ℕ} = { } in order to maintain the empty result. [Each] endsegment which you can name can be removed.

Indeed, Any finite set of endsegments, S, (whether or not S contains elements such that you cannot write down the first element) can be "removed"

>That means the endsegments which cause the empty intersection cannot be defined, not even in principle.

Piffle. "the endsegments which cause the empty intersection",T, is an infinite set of endsegments. T does not have every property that each of its elements has. In particular T cannot be "removed"

--
William Hughes

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, 3 Aug 2022 23:34 UTC

On Wednesday, 3 August 2022 at 18:56:00 UTC-3, WM wrote:
> William schrieb am Mittwoch, 3. August 2022 um 19:36:41 UTC+2:
> > On Wednesday, August 3, 2022 at 12:33:39 PM UTC-3, WM wrote:
>
> > > > > They may have other properties of natural numbers. But we cannot know.
> > > > Of course we can There are many ways, e.g. induction, to know that a number which does not have the property "can be written down", has other properties.
> > > Many.
> > And even if n cannot be written down you "can know" that n has each of these properties.
> Not all, however. The property that a number can be written down (independent of time and ressources) is not a property of these numbers. There is a stronger obstacle than lack of time and ressources.
>
> For instance you cannot name a number n the endsegment E(n) of which cannot removed from ∩{E(k) : k ∈ ℕ} = { } in order to maintain the empty result. Every endsegment which you can name can be removed. That means the endsegments which cause the empty intersection cannot be defined, not even in principle. Why should their other properties be the same?

....as he beats a hasty retreat. You really are a wanker.

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 - Thu, 4 Aug 2022 04:43 UTC

onsdag 3 augusti 2022 kl. 13:00:41 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Mittwoch, 3. August 2022 um 06:52:07 UTC+2:
> > tisdag 2 augusti 2022 kl. 17:31:40 UTC+2 skrev WM:
> > > zelos...@gmail.com schrieb am Montag, 1. August 2022 um 14:34:41 UTC+2:
> > > > måndag 1 augusti 2022 kl. 13:28:50 UTC+2 skrev WM:
> > >
> > > > > > Can you define that set using the usual symbolic language of set theory?
> > > > > The collection is {E(k) | k ∈ ℕ_def /\ ∩{E(1), E(2), ..., E(k) = E(k) /\ |E(k)| = ℵ₀}
> > > > >
> > > > your N_def = N so you have said nothing
> > > If you claim that the intersection of all definable endsegments is empty, then define individually endsegments reducing the infinite intersection to the empty set.
> > >
> > your "define individually" is meaningless. the intersection is empty because n is not in E(n+1)
> >
> > Simple as that. there is nothing else to it
> There is but a minor detail: All definable endsegments appearing in
> ∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀
> can be subtracted from {E(k) : k ∈ ℕ} without changing the result.
>
> ∩({E(k) : k ∈ ℕ} \ {E(k) : k ∈ ℕ_def}) = { }
>
> This proves the existence of dark numbers.
>
> Regards, WM

Nope, all it proves is that an intersection of finite amount of endsegments is different from an INFINITE amount of endsegments intersection.

Which is THE FUCKING POINT!

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, 4 Aug 2022 11:18 UTC

William schrieb am Donnerstag, 4. August 2022 um 00:41:42 UTC+2:
> On Wednesday, August 3, 2022 at 6:56:00 PM UTC-3, WM wrote:
> > The property that a number can be written down (independent of time and ressources) is not a property of these numbers.
> Piffle.
>
> 1 can be written down.
> if n can be written down then n+1 can be written down (independent of time and resources)
>
> Hence each element of |N (without exception) can be written down (independent of time and resources).

No. The elements which index endsegments which have an empty intersection cannot be found, written down, or determined otherwise. There is _no_ way! All definable endsegments can be omitted from
∩{E(k) : k ∈ ℕ} = { } = ∩({E(k) : k ∈ ℕ} \ {E(k) : k ∈ ℕ_def}).
>
> > .. For instance you cannot name a number n the endsegment E(n) of which cannot removed from ∩{E(k) : k ∈ ℕ} = { } in order to maintain the empty result. [Each] endsegment which you can name can be removed.
>
> Indeed, Any finite set of endsegments, S, (whether or not S contains elements such that you cannot write down the first element) can be "removed".

In order to remove a finite set, you have to give the last element.

> >That means the endsegments which cause the empty intersection cannot be defined, not even in principle.
> Piffle. "the endsegments which cause the empty intersection",T, is an infinite set of endsegments. T does not have every property that each of its elements has. In particular T cannot be "removed".

It can be removed collectively. But none of its elements can be defined. They are dark.

Regards, WM

Re: Natural numbers and vases III

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Date: Thu, 4 Aug 2022 04:20:53 -0700 (PDT)
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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Thu, 4 Aug 2022 11:20 UTC

zelos...@gmail.com schrieb am Donnerstag, 4. August 2022 um 06:43:53 UTC+2:
> onsdag 3 augusti 2022 kl. 13:00:41 UTC+2 skrev WM:
> > zelos...@gmail.com schrieb am Mittwoch, 3. August 2022 um 06:52:07 UTC+2:
> > > tisdag 2 augusti 2022 kl. 17:31:40 UTC+2 skrev WM:
> > > > zelos...@gmail.com schrieb am Montag, 1. August 2022 um 14:34:41 UTC+2:
> > > > > måndag 1 augusti 2022 kl. 13:28:50 UTC+2 skrev WM:
> > > >
> > > > > > > Can you define that set using the usual symbolic language of set theory?
> > > > > > The collection is {E(k) | k ∈ ℕ_def /\ ∩{E(1), E(2), ..., E(k) = E(k) /\ |E(k)| = ℵ₀}
> > > > > >
> > > > > your N_def = N so you have said nothing
> > > > If you claim that the intersection of all definable endsegments is empty, then define individually endsegments reducing the infinite intersection to the empty set.
> > > >
> > > your "define individually" is meaningless. the intersection is empty because n is not in E(n+1)
> > >
> > > Simple as that. there is nothing else to it
> > There is but a minor detail: All definable endsegments appearing in
> > ∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀
> > can be subtracted from {E(k) : k ∈ ℕ} without changing the result.
> >
> > ∩({E(k) : k ∈ ℕ} \ {E(k) : k ∈ ℕ_def}) = { }
> >
> > This proves the existence of dark numbers.

> Nope, all it proves is that an intersection of finite amount of endsegments is different from an INFINITE amount of endsegments intersection.

All definable endsegments can be removed without changing the result. Therefore the required endegments are undefinable.
>
> Which is THE FUCKING POINT!

Regards, WM

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, 4 Aug 2022 11:28 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, 4 Aug 2022 11:32 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 - Thu, 4 Aug 2022 11:37 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 - Thu, 4 Aug 2022 11:41 UTC

torsdag 4 augusti 2022 kl. 13:20:59 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Donnerstag, 4. August 2022 um 06:43:53 UTC+2:
> > onsdag 3 augusti 2022 kl. 13:00:41 UTC+2 skrev WM:
> > > zelos...@gmail.com schrieb am Mittwoch, 3. August 2022 um 06:52:07 UTC+2:
> > > > tisdag 2 augusti 2022 kl. 17:31:40 UTC+2 skrev WM:
> > > > > zelos...@gmail.com schrieb am Montag, 1. August 2022 um 14:34:41 UTC+2:
> > > > > > måndag 1 augusti 2022 kl. 13:28:50 UTC+2 skrev WM:
> > > > >
> > > > > > > > Can you define that set using the usual symbolic language of set theory?
> > > > > > > The collection is {E(k) | k ∈ ℕ_def /\ ∩{E(1), E(2), ..., E(k) = E(k) /\ |E(k)| = ℵ₀}
> > > > > > >
> > > > > > your N_def = N so you have said nothing
> > > > > If you claim that the intersection of all definable endsegments is empty, then define individually endsegments reducing the infinite intersection to the empty set.
> > > > >
> > > > your "define individually" is meaningless. the intersection is empty because n is not in E(n+1)
> > > >
> > > > Simple as that. there is nothing else to it
> > > There is but a minor detail: All definable endsegments appearing in
> > > ∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀
> > > can be subtracted from {E(k) : k ∈ ℕ} without changing the result.
> > >
> > > ∩({E(k) : k ∈ ℕ} \ {E(k) : k ∈ ℕ_def}) = { }
> > >
> > > This proves the existence of dark numbers.
> > Nope, all it proves is that an intersection of finite amount of endsegments is different from an INFINITE amount of endsegments intersection.
> All definable endsegments can be removed without changing the result. Therefore the required endegments are undefinable.

Your "definable" is still UNDEFINED!

All you talk about is "removing", "changing", "adding" FINITE stuff. Which is the crux of it all! You think the finite rules holds for infinite, IT DOES NOT!

> >
> > Which is THE FUCKING POINT!
> 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, 4 Aug 2022 11:47 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, 4 Aug 2022 12:12 UTC

On Thursday, August 4, 2022 at 1:47:47 PM UTC+2, WM wrote:

> Definition: An endsegment is called definable, if it can be omitted, together with all its predecessors, from the intersection of all endsegments without changing the result empty set.

CAN YOU PLEEZE USE the usual language used in set theorie, you silly crank?

I guess you mean:

definable(E) :<-> En e IN: E = E(n) & ∩({E(k) : k ∈ ℕ} \ {E(1), ..., E(n)}) = { }

Great. Then we get the result:

AX e {E(k) : k ∈ ℕ}: definable(X)

Ak e IN: definable(E(k))

"All endsegments are definable." (*)

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

Now your claim was:

| "All definable endsegments can be removed without changing the result."

Using the result (*) this means:

| "All endsegments can be removed without changing the result."

Well... AGAIN I HAVE TO ASK: CAN YOU PLEEZE USE the usual language used in set theorie, you silly crank?

Holy shit!

Hint:

TRUE: Ak e IN: ∩({E(k) : k ∈ ℕ} \ {E(k)}) = { }

WRONG: Ak e IN: ∩({E(k) : k ∈ ℕ} \ {E(k) : k ∈ ℕ}) = { }.

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, 4 Aug 2022 12:13 UTC

On Thursday, August 4, 2022 at 1:47:47 PM UTC+2, WM wrote:

> Definition: An endsegment is called definable, if it can be omitted, together with all its predecessors, from the intersection of all endsegments without changing the result empty set.

CAN YOU PLEEZE USE the usual language used in set theory, you silly crank?

I guess you mean:

definable(E) :<-> En e IN: E = E(n) & ∩({E(k) : k ∈ ℕ} \ {E(1), ..., E(n)}) = { }

Great. Then we get the result:

AX e {E(k) : k ∈ ℕ}: definable(X)

Ak e IN: definable(E(k))

"All endsegments are definable." (*)

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

Now your claim was:

| "All definable endsegments can be removed without changing the result."

Using the result (*) this means:

| "All endsegments can be removed without changing the result."

Well... AGAIN I HAVE TO ASK: CAN YOU PLEEZE USE the usual language used in set theory, you silly crank?

Holy shit!

Hint:

TRUE: Ak e IN: ∩({E(k) : k ∈ ℕ} \ {E(k)}) = { }

WRONG: Ak e IN: ∩({E(k) : k ∈ ℕ} \ {E(k) : k ∈ ℕ}) = { }.

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, 4 Aug 2022 12:30 UTC

On Thursday, 4 August 2022 at 08:18:43 UTC-3, WM wrote:
> William schrieb am Donnerstag, 4. August 2022 um 00:41:42 UTC+2:
> > On Wednesday, August 3, 2022 at 6:56:00 PM UTC-3, WM wrote:
> > > The property that a number can be written down (independent of time and ressources) is not a property of these numbers.
> > Piffle.
> >
> > 1 can be written down.
> > if n can be written down then n+1 can be written down (independent of time and resources)
> >
> > Hence each element of |N (without exception) can be written down (independent of time and resources).
> No. The elements which index endsegments which have an empty intersection cannot be found, written down, or determined otherwise. There is _no_ way! All definable endsegments can be omitted from
> ∩{E(k) : k ∈ ℕ} = { } = ∩({E(k) : k ∈ ℕ} \ {E(k) : k ∈ ℕ_def}).
> >
> > > .. For instance you cannot name a number n the endsegment E(n) of which cannot removed from ∩{E(k) : k ∈ ℕ} = { } in order to maintain the empty result. [Each] endsegment which you can name can be removed.
> >
> > Indeed, Any finite set of endsegments, S, (whether or not S contains elements such that you cannot write down the first element) can be "removed".
>
> In order to remove a finite set, you have to give the last element.
> > >That means the endsegments which cause the empty intersection cannot be defined, not even in principle.
> > Piffle. "the endsegments which cause the empty intersection",T, is an infinite set of endsegments. T does not have every property that each of its elements has. In particular T cannot be "removed".
>
> It can be removed collectively. But none of its elements can be defined. They are dark.

Come on, Muckenheim. It seems you are now finally completely brain dead.
Let E be the set of even natural numbers and O be the set of odd natural numbers.
Then ∩({E(k) : k ∈ E} = ∩({E(k) : k ∈ O} = {}. Since E ∩ O = {} as well, this shows that *NO* natural number in and of itself has any influence on the intersection of the remaining end segments.

Re: Natural numbers and vases III

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From: FTR...@nomail.afraid.org (FromTheRafters)
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Subject: Re: Natural numbers and vases III
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by: FromTheRafters - Thu, 4 Aug 2022 12:57 UTC

Re: Natural numbers and vases III

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by: FromTheRafters - Thu, 4 Aug 2022 13:05 UTC

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: monteu...@t-online.de (WM)
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by: WM - Thu, 4 Aug 2022 13:40 UTC

Re: Natural numbers and vases III

<a56c4c39-812b-4127-a720-2d1d06f7191cn@googlegroups.com>

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Subject: Re: Natural numbers and vases III
From: monteu...@t-online.de (WM)
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by: WM - Thu, 4 Aug 2022 13:41 UTC

Re: Natural numbers and vases III

<643d6709-a2f6-4cfc-8ef2-048f43b88608n@googlegroups.com>

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Subject: Re: Natural numbers and vases III
From: monteu...@t-online.de (WM)
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by: WM - Thu, 4 Aug 2022 13:43 UTC

Re: Natural numbers and vases III

<07dd58b2-5812-441e-bb3c-f96d3ee8f185n@googlegroups.com>

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https://www.novabbs.com/tech/article-flat.php?id=108124&group=sci.math#108124

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Subject: Re: Natural numbers and vases III
From: monteu...@t-online.de (WM)
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by: WM - Thu, 4 Aug 2022 13:47 UTC

Re: Natural numbers and vases III

<4c6f2df7-34e5-4325-b4b6-682a0eaebed9n@googlegroups.com>

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Subject: Re: Natural numbers and vases III
From: wpihug...@gmail.com (William)
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by: William - Thu, 4 Aug 2022 14:08 UTC

Re: Natural numbers and vases III

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https://www.novabbs.com/tech/article-flat.php?id=108127&group=sci.math#108127

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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, 4 Aug 2022 14:22 UTC

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