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

Re: Natural numbers and vases

<t78r0l$1hsr$2@gioia.aioe.org>

https://www.novabbs.com/tech/article-flat.php?id=101784&group=sci.math#101784

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From: inva...@invalid.com (sergi o)
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Subject: Re: Natural numbers and vases
Date: Wed, 1 Jun 2022 17:59:33 -0500
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by: sergi o - Wed, 1 Jun 2022 22:59 UTC

Re: Natural numbers and vases

<t78r3e$1hsr$3@gioia.aioe.org>

copy mid

https://www.novabbs.com/tech/article-flat.php?id=101785&group=sci.math#101785

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From: inva...@invalid.com (sergi o)
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Subject: Re: Natural numbers and vases
Date: Wed, 1 Jun 2022 18:01:02 -0500
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by: sergi o - Wed, 1 Jun 2022 23:01 UTC

Re: Natural numbers and vases

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https://www.novabbs.com/tech/article-flat.php?id=101891&group=sci.math#101891

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Fri, 3 Jun 2022 13:13 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 - Fri, 3 Jun 2022 13:32 UTC

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
Injection-Date: Fri, 03 Jun 2022 13:34:55 +0000
Content-Type: text/plain; charset="UTF-8"

by: WM - Fri, 3 Jun 2022 13:34 UTC

William schrieb am Donnerstag, 2. Juni 2022 um 00:23:44 UTC+2:
> On Wednesday, June 1, 2022 at 5:28:26 PM UTC-3, WM wrote:
> > William schrieb am Mittwoch, 1. Juni 2022 um 22:13:10 UTC+2:
> > > [The state "all fractions are enumerated" ] is demonstrated by showing that a set of ordered pairs exists (A bijectiion is a set).
> > It is contradicted by showing that the set of ordered pairs does not exist.
> >
> > This is a contradiction.
> Piffle. It is a set of an matrices. It says nothing about the existence of a set of ordered pairs;
Cantor's formula
k = (m + n - 1)(m + n - 2)/2 + m
defines the terms of the infinite sequence 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ... .
The terms of this very infinite sequence can also be written as matrices:
XOOO...
XOOO...
XOOO...
XOOO...
....

XXOO...
OOOO...
XOOO...
XOOO...
....

XXOO...
XOOO...
OOOO...
XOOO...
....

XXXO...
XOOO...
OOOO...
OOOO...
....

and on and on
where the underlying matrix contains all positive fractions

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
5/1, 5/2, 5/3, 5/4, ...
....

The X are indeXes and the O are vOids.

This is a bit more tedious and it requires more space (which is cheap here) but it is precisely a representation of Cantor's ordered pairs, and it has the big advantage to show that never, in infinity, all fractions will have been indexed.

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
Injection-Date: Fri, 03 Jun 2022 14:24:17 +0000
Content-Type: text/plain; charset="UTF-8"

by: William - Fri, 3 Jun 2022 14:24 UTC

Re: Natural numbers and vases

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https://www.novabbs.com/tech/article-flat.php?id=101898&group=sci.math#101898

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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by: William - Fri, 3 Jun 2022 14:35 UTC

On Friday, June 3, 2022 at 10:35:01 AM UTC-3, WM wrote:
> William schrieb am Donnerstag, 2. Juni 2022 um 00:23:44 UTC+2:
> > On Wednesday, June 1, 2022 at 5:28:26 PM UTC-3, WM wrote:
> > > William schrieb am Mittwoch, 1. Juni 2022 um 22:13:10 UTC+2:
> > > > [The state "all fractions are enumerated" ] is demonstrated by showing that a set of ordered pairs exists (A bijectiion is a set).
> > > It is contradicted by showing that the set of ordered pairs does not exist.
> > >
> > > This is a contradiction.
> > Piffle. It is a set of an matrices. It says nothing about the existence of a set of ordered pairs;
> Cantor's formula
> k = (m + n - 1)(m + n - 2)/2 + m
> defines the terms of the infinite sequence 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ... .
> The terms of this very infinite sequence can also be written as matrices:

Indeed, it is the set of matrices that can be reached by doing exchanges

> XOOO...
> XOOO...
> XOOO...

> XOOO...
> ...
>
> XXOO...
> OOOO...
> XOOO...
> XOOO...
> ...
>
> XXOO...
> XOOO...
> OOOO...
> XOOO...
> ...
>
> XXXO...
> XOOO...
> OOOO...
> OOOO...
> ...

What you have shown is that the state "all exchanges have been done"

>
> 1/1, 1/2, 1/3, 1/4, ...
> 2/1, 2/2, 2/3, 2/4, ...
> 3/1, 3/2, 3/3, 3/4, ...
> 4/1, 4/2, 4/3, 4/4, ...
> 5/1, 5/2, 5/3, 5/4, ...
> ...
>
is not reached by doing exchanges, but does exist. So there is no contradiction in saying it exists

--
William Hughes

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, 4 Jun 2022 14:08 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, 4 Jun 2022 14:15 UTC

Re: Natural numbers and vases

<87c806c6-30dc-459f-8e5f-da1243b2a80cn@googlegroups.com>

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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by: William - Sat, 4 Jun 2022 14:50 UTC

Re: Natural numbers and vases

<1a5a898b-dc9c-4948-9aa6-3231dcf06c15n@googlegroups.com>

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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by: William - Sat, 4 Jun 2022 14:53 UTC

Re: Natural numbers and vases

<9ec06c08-4dc9-43cb-8ee3-1ddef1f7ce5bn@googlegroups.com>

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Sun, 5 Jun 2022 13:58 UTC

Re: Natural numbers and vases

<237da8ab-8bec-489c-bbbc-c1a43d64bf6an@googlegroups.com>

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Sun, 5 Jun 2022 14:06 UTC

William schrieb am Samstag, 4. Juni 2022 um 16:54:03 UTC+2:
> On Saturday, June 4, 2022 at 11:08:15 AM UTC-3, WM wrote:
> > William schrieb am Freitag, 3. Juni 2022 um 16:24:23 UTC+2:
> > > On Friday, June 3, 2022 at 10:32:57 AM UTC-3, WM wrote:
> > >
> > > > ... your claim is a self-contradiction: The state "all exchanges are done" is not reached by doing exchanges.
> > >
> > > This is not a self-contradiction, thought it is non--intuitive.
> > By definition
> Piffle. The state "all exchanges have been done" has no intrinsic meaning.. If it refers to anything it refers to a limit.

Never. It refers to the indexing of all positive rationals.

> Note that the limit of a sequence can have a property that no element of the sequence has.

Counting must be done. It is not a matter of limits. Limits usually do not belong to the sequence. But counting means to issue indices such that every pair can be found within the order of the sequence. Limits could be claimed but would never prove the indexing,

> You do not like limits that do not involve things getting closer.

These are not limits but nonsense.

> Fine, then "all exchanges have been done" has no meaning. The statement "there is a bijection from the naturals to the positive rationals" does not require "all exchanges have been done" to have a meaning.

In my model, which is an exact copy of Cantor's theory, just this is required. "All exchanges have been done" is tantamount to "all indices have been issued". If this is impossible or only possible "in the limit" whatever that should mean, then it shows that Cantor's theory fails.

PS: In the limit also the diagonal number can be a member of the list. This is as easy to prove or to disprove as the enumerating of all fractions.

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 - Sun, 5 Jun 2022 14:40 UTC

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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by: William - Sun, 5 Jun 2022 14:48 UTC

Re: Natural numbers and vases

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https://www.novabbs.com/tech/article-flat.php?id=102053&group=sci.math#102053

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Sun, 5 Jun 2022 16:22 UTC

Re: Natural numbers and vases

<90f11133-eb99-45c1-bf95-17b5d95c8eb7n@googlegroups.com>

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https://www.novabbs.com/tech/article-flat.php?id=102055&group=sci.math#102055

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Sun, 5 Jun 2022 16:36 UTC

Re: Natural numbers and vases

<94779ba4-ee8c-43eb-86c8-44b2a253d2e2n@googlegroups.com>

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https://www.novabbs.com/tech/article-flat.php?id=102062&group=sci.math#102062

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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by: William - Sun, 5 Jun 2022 18:38 UTC

Re: Natural numbers and vases

<t7jj4l$sj5$1@dont-email.me>

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From: chris.m....@gmail.com (Chris M. Thomasson)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
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by: Chris M. Thomasson - Mon, 6 Jun 2022 00:52 UTC

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Mon, 6 Jun 2022 04:10 UTC

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Mon, 6 Jun 2022 16:21 UTC

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
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by: WM - Mon, 6 Jun 2022 16:23 UTC

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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by: William - Mon, 6 Jun 2022 17:31 UTC

Re: Natural numbers and vases

<2bf5abd0-5dd7-681e-66ee-a949d7d8fdc7@att.net>

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https://www.novabbs.com/tech/article-flat.php?id=102124&group=sci.math#102124

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Mon, 6 Jun 2022 13:39:23 -0400
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by: Jim Burns - Mon, 6 Jun 2022 17:39 UTC

On 6/6/2022 12:23 PM, WM wrote:
> Chris M. Thomasson schrieb
> am Montag, 6. Juni 2022 um 02:52:47 UTC+2:
>> On 6/5/2022 6:58 AM, WM wrote:

>>>> Incorrect. Correct is that state of
>>>> "all fraction having been indexed"
>>>> cannot be accomplished by doing exchanges.
>>>
>>> If doing is not allowed,
>>> than "having been done" is impossible.
>>
>> There is no "having been done"
>> in an infinite process... :^)
>
> Then Cantor's claim is unattainable:
> Completion, element by element.

This is attainable:

(i)
Make a claim which we know to be true of
each one of infinitely-many individuals --
because we know what we say means.
| | If k-1 exists, then k+1 exists.
| | For each _finite_ ⟨1,...,k⟩,
| for each BEFORE and AFTER ⊂ ⟨1,...,k⟩,
| some j ends BEFORE and j+1 begins AFTER.

We can call claims like these _complete_
in the sense that no individual which we are
currently discussing is excluded.

(ii)
Make further claims which we also know to
be true of those same individuals --
because we only proceed to a further claim
by steps which preserve the uniform truth
of earlier claims.
| | For each fraction m/n with finite ⟨1,...,m⟩
| and finite ⟨1,...,n⟩,
| there is a unique index k = (m+n-1)*(m+n-2)/2+m
| with finite ⟨1,...,k⟩.
| | Some collections can match a proper subset.

These further claims are _complete_ in the same
sense: no individual which we are currently
discussing is excluded.

This sense of completeness propels us to Cantor's
Paradise, but non-mystically. It doesn't take
a miracle to make a claim true of infinitely-many.
It doesn't take a miracle to NOT make claims
that DON'T preserve that uniform truth.

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
Injection-Date: Mon, 06 Jun 2022 18:17:55 +0000
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by: WM - Mon, 6 Jun 2022 18:17 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.