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

Re: Natural numbers and vases

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https://www.novabbs.com/tech/article-flat.php?id=105122&group=sci.math#105122

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From: inva...@invalid.com (sergi o)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Fri, 1 Jul 2022 09:02:59 -0500
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by: sergi o - Fri, 1 Jul 2022 14:02 UTC

Re: Natural numbers and vases

<t9mvcq$7b2$1@gioia.aioe.org>

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From: inva...@invalid.com (sergi o)
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Subject: Re: Natural numbers and vases
Date: Fri, 1 Jul 2022 09:12:41 -0500
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by: sergi o - Fri, 1 Jul 2022 14:12 UTC

On 7/1/2022 6:09 AM, WM wrote:
> Jim Burns schrieb am Freitag, 1. Juli 2022 um 02:47:25 UTC+2:
>> On 6/29/2022 1:45 PM, WM wrote:
>>
>>> The Peano set has all last numbers
>>> but no last last number.
>> The Peano set contains one and only one subset (itself)
>> which is closed under k --> k+1 and contains 0.
>
> Every k∈|N has ℵo not yet distinguished successors. When and where will all have been distinguished?

you need to distinguish the last k first.

>>
>> Thus,
>> if k is in S, then k+1 is in S
>
> If k has ℵo not yet distinguished successors, then k+1 has ℵo not yet distinguished successors.

No, as k would have k+1 as a "distinguished" successors.

by the way, what is your definition of "distinguished" ? It is not a term used in math at all.

> When will all of them be distinguished?

as soon as you get to the last one.

>>
>> S is closed under k --> k+1
>>
>> ... _and_ S contains 0
>>
>> Therefore,
>> the only subset which S can be is the whole Peano set.
>

>
> Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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by: William - Fri, 1 Jul 2022 15:31 UTC

Re: Natural numbers and vases

<629c3bc3-ca47-8cfc-cdcb-07ec048ab822@att.net>

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From: james.g....@att.net (Jim Burns)
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Subject: Re: Natural numbers and vases
Date: Fri, 1 Jul 2022 12:16:12 -0400
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by: Jim Burns - Fri, 1 Jul 2022 16:16 UTC

On 7/1/2022 7:09 AM, WM wrote:
> Jim Burns schrieb
> am Freitag, 1. Juli 2022 um 02:47:25 UTC+2:
>> On 6/29/2022 1:45 PM, WM wrote:

>>> The Peano set has all last numbers
>>> but no last last number.
>>
>> The Peano set contains one and only one subset (itself)
>> which is closed under k --> k+1 and contains 0.
>
> Every k∈|N has ℵo not yet distinguished successors.

Each k ∈ ℕ has |ℕ|-many successors,
distinguished or not,
each of which has fewer than |ℕ|-many predecessors,
distinguished or not.

Infinity is not
a reallyreallyreallyreallyreallyreally
large number.

> When and where will all have been distinguished?

We can make descriptive claims of k ∈ ℕ.
They will be true of each k ∈ ℕ, distinguished or not.

We can extend the claims we make, allowing only
claims that must be true when earlier claims are true.
Because we start with claims that are true of each
k ∈ ℕ, distinguished or not, later claims are
true of each k ∈ ℕ, distinguished or not.

The claims we make about k ∈ ℕ are as unaffected by
the status, distinguished-or-not, of any k ∈ ℕ
as they are unaffected by whether any k ∈ ℕ tastes
like chocolate or collects bubble-gum cards.

I don't know how to answer, but it doesn't matter.

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
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by: WM - Sat, 2 Jul 2022 14:49 UTC

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: zelos.ma...@gmail.com (zelos...@gmail.com)
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by: zelos...@gmail.com - Sat, 2 Jul 2022 15:34 UTC

torsdag 30 juni 2022 kl. 14:54:29 UTC+1 skrev WM:
> William schrieb am Donnerstag, 30. Juni 2022 um 01:25:26 UTC+2:
> > |N_def is a set. A set does not change.
> ℕ_def is many sets, depending on what elements are defined. It is a collection.

no, it is A set, a singular set that never changes. And N is a subset of it..

> Compare the set of known prime numbers. It is a set now but another set next year. It is a potentially infinite collection.

Set P_Y2022 and set P_Y2023 may very well be different sets, but they are DIFFERENT SETS STILL!

It is not A set that changes!

>
> Further, if ℕ was endless, then ∩{E(k) : k ∈ ℕ} could not be empty, because every endsegment which is not the last one cannot make the intersection empty because there is always another endsegment reducing the intersection further. This would not be possible if the intersection was empty already.

this is false on its own in so many ways.

There is no step by step process. The intersection asks a simple question, which element is in all the sets given?
Because NO element is that, the intersection has NO elements.
>
> Conclusion: An endless set of endsegments cannot have an empty intersection ut only a shrinking one, forever.
>
> Regards, WM

Re: Natural numbers and vases

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by: sergi o - Sat, 2 Jul 2022 16:01 UTC

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Sat, 2 Jul 2022 16:25 UTC

William schrieb am Freitag, 1. Juli 2022 um 17:31:53 UTC+2:
> On Friday, July 1, 2022 at 8:01:01 AM UTC-3, WM wrote:
> > William schrieb am Freitag, 1. Juli 2022 um 00:24:51 UTC+2:
> > > On Thursday, June 30, 2022 at 6:51:30 PM UTC-3, WM wrote:
> > > > William schrieb am Donnerstag, 30. Juni 2022 um 22:14:43 UTC+2:
> >
> > > > > Piffle. This is true by simple induction on the Peano set |N. Recall induction is valid ∀k∈N
> > > > only for the finite collection
> > > Piffle. Induction is valid ∀k∈|N
> > [Each] k∈|N has ℵo not yet [written down] successors.
>
> So what if they have not been written down? ( "distinguished" is one of you many many many many ways of saying "has been written down"). Despite the misleading term it is not necessary to write down two elements of |N to know that they are different.

But there are elements that cannot be distiguished:

For all definable numbers we have the intersection of their endsegments
|∩{E(k) : k ∈ ℕ_def}| = ℵ₀

For all natural numbers, we have the empty intersection of their endsegments:
∩{E(k) : k ∈ ℕ} = { }

The endsegments can only shrink by on number per endsegment
∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}

What natural numbers reduce the intersection from ℵ₀ to zero?

They cannot be distinguished. You don't like this fact and don't comment, but it is a fact.

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Sat, 2 Jul 2022 16:34 UTC

zelos...@gmail.com schrieb am Samstag, 2. Juli 2022 um 17:34:28 UTC+2:
> torsdag 30 juni 2022 kl. 14:54:29 UTC+1 skrev WM:
> > William schrieb am Donnerstag, 30. Juni 2022 um 01:25:26 UTC+2:
> > > |N_def is a set. A set does not change.
> > ℕ_def is many sets, depending on what elements are defined. It is a collection.
> no, it is A set, a singular set that never changes. And N is a subset of it.
>
> > Compare the set of known prime numbers. It is a set now but another set next year. It is a potentially infinite collection.
>
> Set P_Y2022 and set P_Y2023 may very well be different sets, but they are DIFFERENT SETS STILL!
>
> It is not A set that changes!

It is what we call a potentially infinite collection.
>
> >
> > Further, if ℕ was endless, then ∩{E(k) : k ∈ ℕ} could not be empty, because every endsegment which is not the last one cannot make the intersection empty because there is always another endsegment reducing the intersection further. This would not be possible if the intersection was empty already.
>
> this is false on its own in so many ways.

It is fact.
>
> There is no step by step process. The intersection asks a simple question, which element is in all the sets given?

For all definable numbers k we can obtain the process step by step and find the correct result:
∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}

> Because NO element is that, the intersection has NO elements.

Correct.
∩{E(k) : k ∈ ℕ} = { }
But for all definable elements we obtain
|∩{E(k) : k ∈ ℕ_def}| = ℵ₀

If the rest is no step by step process, then the only reason can be dark numbers.

Regards, WM

Re: Natural numbers and vases

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by: sergi o - Sat, 2 Jul 2022 16:34 UTC

On 7/2/2022 11:25 AM, WM wrote:
> William schrieb am Freitag, 1. Juli 2022 um 17:31:53 UTC+2:
>> On Friday, July 1, 2022 at 8:01:01 AM UTC-3, WM wrote:
>>> William schrieb am Freitag, 1. Juli 2022 um 00:24:51 UTC+2:
>>>> On Thursday, June 30, 2022 at 6:51:30 PM UTC-3, WM wrote:
>>>>> William schrieb am Donnerstag, 30. Juni 2022 um 22:14:43 UTC+2:
>>>
>>>>>> Piffle. This is true by simple induction on the Peano set |N. Recall induction is valid ∀k∈N
>>>>> only for the finite collection
>>>> Piffle. Induction is valid ∀k∈|N
>>> [Each] k∈|N has ℵo not yet [written down] successors.
>>
>> So what if they have not been written down? ( "distinguished" is one of you many many many many ways of saying "has been written down"). Despite the misleading term it is not necessary to write down two elements of |N to know that they are different.
>
> But there are elements that cannot be distiguished:
>
> For all definable numbers we have the intersection of their endsegments
> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀

"_def" is meaningless

>
> For all natural numbers, we have the empty intersection of their endsegments:
> ∩{E(k) : k ∈ ℕ} = { }

finally an admission that the intersection of all endsegments is empty.

>
> The endsegments can only shrink by on number per endsegment
> ∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}

Wrong. Sets do not shrink. Also you stop at k.

>
> What natural numbers reduce the intersection from ℵ₀ to zero?

Wrong. the intersection is not an Endsegment

>
> They cannot be distinguished.

wrong, you cannot read and understand equations.

You don't like this fact and don't comment, but it is a fact.

no, it is a spoof. Your silly spoof.

>
> Regards, WM

Re: Natural numbers and vases

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by: WM - Sat, 2 Jul 2022 16:36 UTC

Re: Natural numbers and vases

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Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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by: William - Sat, 2 Jul 2022 17:06 UTC

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
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by: William - Sat, 2 Jul 2022 17:21 UTC

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Sat, 2 Jul 2022 17:27 UTC

Re: Natural numbers and vases

<b295f9a1-bc23-4c99-8826-576ddc640e98n@googlegroups.com>

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Newsgroups: sci.math
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Subject: Re: Natural numbers and vases
From: zelos.ma...@gmail.com (zelos...@gmail.com)
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by: zelos...@gmail.com - Sat, 2 Jul 2022 17:31 UTC

lördag 2 juli 2022 kl. 17:25:15 UTC+1 skrev WM:
> William schrieb am Freitag, 1. Juli 2022 um 17:31:53 UTC+2:
> > On Friday, July 1, 2022 at 8:01:01 AM UTC-3, WM wrote:
> > > William schrieb am Freitag, 1. Juli 2022 um 00:24:51 UTC+2:
> > > > On Thursday, June 30, 2022 at 6:51:30 PM UTC-3, WM wrote:
> > > > > William schrieb am Donnerstag, 30. Juni 2022 um 22:14:43 UTC+2:
> > >
> > > > > > Piffle. This is true by simple induction on the Peano set |N. Recall induction is valid ∀k∈N
> > > > > only for the finite collection
> > > > Piffle. Induction is valid ∀k∈|N
> > > [Each] k∈|N has ℵo not yet [written down] successors.
> >
> > So what if they have not been written down? ( "distinguished" is one of you many many many many ways of saying "has been written down"). Despite the misleading term it is not necessary to write down two elements of |N to know that they are different.
> But there are elements that cannot be distiguished:

distinguished has no meaning in mathematics

>
> For all definable numbers we have the intersection of their endsegments
> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀

false, you keep assuming this but it is false.

>
> For all natural numbers, we have the empty intersection of their endsegments:
> ∩{E(k) : k ∈ ℕ} = { }

given that N c N_def , you should understand that both are empty.

>
> The endsegments can only shrink by on number per endsegment
> ∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
>
> What natural numbers reduce the intersection from ℵ₀ to zero?

there is no single specicifc one, how hard is that to fucking understand`

>
> They cannot be distinguished. You don't like this fact and don't comment, but it is a fact.

This is not a fact, this is an empty pointless stupid assertion by you with no mathematical merit

>
> Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Sat, 2 Jul 2022 17:33 UTC

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Sat, 2 Jul 2022 17:37 UTC

zelos...@gmail.com schrieb am Samstag, 2. Juli 2022 um 19:31:34 UTC+2:
> lördag 2 juli 2022 kl. 17:25:15 UTC+1 skrev WM:
> >
> > For all definable numbers we have the intersection of their endsegments
> > |∩{E(k) : k ∈ ℕ_def}| = ℵ₀
> false, you keep assuming this but it is false.

I defined ℕ_def by this property.
> >
> > For all natural numbers, we have the empty intersection of their endsegments:
> > ∩{E(k) : k ∈ ℕ} = { }
> given that N c N_def , you should understand that both are empty.
> >
> > The endsegments can only shrink by on number per endsegment
> > ∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
> >
> > What natural numbers reduce the intersection from ℵ₀ to zero?
> there is no single specicifc one, how hard is that to fucking understand`

According to
∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
there are single specific endsegments.
> >
> > They cannot be distinguished. You don't like this fact and don't comment, but it is a fact.
> This is not a fact, this is an empty pointless stupid assertion by you with no mathematical merit

No merits are required to observe mathematical rules.

Regards, WM

Re: Natural numbers and vases

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by: William - Sat, 2 Jul 2022 17:47 UTC

Re: Natural numbers and vases

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Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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by: William - Sat, 2 Jul 2022 18:14 UTC

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: horand.g...@gmail.com (Gus Gassmann)
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by: Gus Gassmann - Sat, 2 Jul 2022 19:06 UTC

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