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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 - Sat, 28 May 2022 00:27 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 - Sat, 28 May 2022 00:34 UTC

Re: Natural numbers and vases

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Sat, 28 May 2022 00:31:00 -0400
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by: Jim Burns - Sat, 28 May 2022 04:31 UTC

On 5/27/2022 8:27 PM, Fritz Feldhase wrote:
> On Saturday, May 28, 2022 at 1:14:08 AM UTC+2,
> Jim Burns wrote

>> When all swaps have been swapped,
>
> There is no such state in mathematics,
> you silly idiot!

I already defined for you what I mean by
"When all swaps have been swapped,"
There are NO INFINITELY-DISTANT swaps.

If Bob is not at place p
just before FINITELY-DISTANT swap j
and
no later FINITELY-DISTANT swap k >= j
swaps Bob into p
then BY DEFINITION, that's what I mean by
when all swaps have been swapped,
Bob is not at p.

And while we're doing this...

If Bob is at place p
just before FINITELY-DISTANT swap j
and
no later FINITELY-DISTANT swap k >= j
swaps Bob out of p
then BY DEFINITION, that's what I mean by
when all swaps have been swapped,
Bob is at p.

> WE DO NOT ARGUE BY REFERRING TO SUPERTASKS
> in set theory/mathematics.

We do not pretend that things have been defined
differently from how they've been defined
in set theory/mathematics.

> Hell, you are just as cranky as your "opponent" Mückenheim.
>
> Holy shit!
>
>> no O is left.
>> No O is left in a place with a swap in and a swap out.
>> No O is left in a place with no swap in.
>>> Do you agree that in finite distance from start
>>> the 1:1 parity is guaranteed?
>> There is no finite distance in which
>> every exchange of X and O happens.

Re: Natural numbers and vases

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by: FromTheRafters - Sat, 28 May 2022 09:14 UTC

Re: Natural numbers and vases

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by: sergi o - Sat, 28 May 2022 13:13 UTC

Re: Natural numbers and vases

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by: sergi o - Sat, 28 May 2022 13:15 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, 28 May 2022 19:10 UTC

William schrieb am Freitag, 27. Mai 2022 um 22:16:21 UTC+2:
> On Friday, May 27, 2022 at 4:50:10 PM UTC-3, WM wrote:
> > William schrieb am Freitag, 27. Mai 2022 um 21:41:04 UTC+2:
> > > On Friday, May 27, 2022 at 4:31:58 PM UTC-3, WM wrote:
> > > > William schrieb am Donnerstag, 26. Mai 2022 um 19:00:49 UTC+2:
> > > > > > ... the *process* of exchanges is fully symmetric. An X leaves its place and occupies the place of an O
> > > > >
> > > > > Piffle. The place that the X leaves and the place the the O leaves differ. Thus this is not "fully symmetric".
> > > > >
> > > > By the difference of place-numbers a deviation from the 1:1 parity can't happen, can it?
> > > Piffle. You said it was "fully symmetric".
> > By symmetry I addressed this: If the exchange of one X and one O can result in two X and no O,
> It cannot.

Then all prattle about the sets is irrelevant.

> Before the exchange there is a set of coordintates of the X's with cardinality aleph_0 and a set of coordinates of the O'x with cardinality aleph_0.

My result shows that cardinality is an invalid concept.

> Performing the exchange does not change the cardinalities, but does change the sets.

If cardinality was a meaningful concept, performing all exchanges should change the cardinalities.

> Since the sets and the changes to the sets are different the process is not "fully symmetric".

Only with respect to parity conservation of X and O.
>
> You have shown a result that is strongly counterintuitive. You have not shown a contradiction/

My result is clear and not counterintuitve. Cantor's claim is wrong.

Regards, WM

Re: Natural numbers and vases

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by: sergi o - Sat, 28 May 2022 19:11 UTC

Re: Natural numbers and vases

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by: WM - Sat, 28 May 2022 19:17 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, 28 May 2022 19:22 UTC

Fritz Feldhase schrieb am Samstag, 28. Mai 2022 um 02:34:08 UTC+2:
> On Friday, May 27, 2022 at 9:34:28 PM UTC+2, WM wrote:

> > Nevertheless every exchange of X and O happens in finite distance from the start.
> Yes, you are right.
>
> Hint: Jim Beam's chess examply is quite helpful for pointing out a fundamental error in your "argument". There are positions in chess which cannot be reached by any number of regular moves (starting with the initial position). Still this does not mean that these positions do not exist!

But it means that they cannot be reached. Cantor claims that the complete indexing can be reached.
>
> In the same way there *is* a matrix containing just the fractions 1/1, 2/1, 3/1, ...

Of course, but it is much smaller than the matrix containing all fractions.

> though it cannot be "reached" by any number of swaps.

Whatever. Cantor claims that the complete indexing can be performed: "Wenn zwei wohldefinierte Mannigfaltigkeiten M und N sich eindeutig und vollständig, Element für Element, einander zuordnen lassen (was, wenn es auf eine Art möglich ist, immer auch noch auf viele andere Weisen geschehen kann), so möge für das Folgende die Ausdrucksweise gestattet sein, daß diese Mannigfaltigkeiten gleiche Mächtigkeit haben, oder auch, daß sie äquivalent sind." [Cantor, p. 119]

Not if we have to jump from an existing state to a not existing delusion.

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, 28 May 2022 19:52 UTC

Jim Burns schrieb am Samstag, 28. Mai 2022 um 01:14:08 UTC+2:
> On 5/27/2022 3:34 PM, WM wrote:
> > Jim Burns schrieb
> > am Donnerstag, 26. Mai 2022 um 18:38:43 UTC+2:
>
> >> Infinity is not a reallyreallyreallyreallyreallyreally
> >> large number. It is a different kind of thing.
> >
> > Nevertheless every exchange of X and O happens in
> > finite distance from the start.
> There is no finite distance in which
> every exchange of X and O happens.

Is there a first infinite distance? No.
But there is no finite border because every finite broder can be surpassed. But all swaps happen in finite distance each. We call that potential infinity.
>
> Each place with a swap in has a swap out.

This does never end and is never stopped by omega or lack of places.

> This is possible because
> there is no finite distance in which
> every exchange of X and O happens.

And there is no infinite distance and no completeness.
>
> When all the swaps have swapped,
> no O is left in a place with a swap out.
> This is possible because
> each place with a swap in has a swap out.
>
> When all swaps have been swapped,

then there would be completeness.

> no O is left.

That is a contradiction because never an O is lost. O in and O out is basic.

> No O is left in a place with a swap in and a swap out.
> No O is left in a place with no swap in.
> > Do you agree that in finite distance from start
> > the 1:1 parity is guaranteed?
> There is no finite distance in which
> every exchange of X and O happens.

For each place with an O out there is a place with an O in. Never an O gets lost. And that is what counts.

Regards, WM

Re: Natural numbers and vases

<t6u5jf$5hs$1@gioia.aioe.org>

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Newsgroups: sci.math
Subject: Re: Natural numbers and vases
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by: sergi o - Sat, 28 May 2022 21:52 UTC

Re: Natural numbers and vases

<t6u5p3$96b$1@gioia.aioe.org>

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From: inva...@invalid.com (sergi o)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Sat, 28 May 2022 16:55:46 -0500
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by: sergi o - Sat, 28 May 2022 21:55 UTC

On 5/28/2022 2:52 PM, WM wrote:
> Jim Burns schrieb am Samstag, 28. Mai 2022 um 01:14:08 UTC+2:
>> On 5/27/2022 3:34 PM, WM wrote:
>>> Jim Burns schrieb
>>> am Donnerstag, 26. Mai 2022 um 18:38:43 UTC+2:
>>
>>>> Infinity is not a reallyreallyreallyreallyreallyreally
>>>> large number. It is a different kind of thing.
>>>
>>> Nevertheless every exchange of X and O happens in
>>> finite distance from the start.
>> There is no finite distance in which
>> every exchange of X and O happens.
>
> Is there a first infinite distance? No.
> But there is no finite border because every finite broder can be surpassed. But all swaps happen in finite distance each. We call that potential infinity.

no, we call that bullshit.

>>
>> Each place with a swap in has a swap out.
>
> This does never end and is never stopped by omega or lack of places.

"swap" is not a math operation.

>
>> This is possible because
>> there is no finite distance in which
>> every exchange of X and O happens.
>
> And there is no infinite distance and no completeness.

that is meaningless.

>>
>> When all the swaps have swapped,
>> no O is left in a place with a swap out.
>> This is possible because
>> each place with a swap in has a swap out.
>>
>> When all swaps have been swapped,
>
> then there would be completeness.

show your work.

>
>> no O is left.
>
> That is a contradiction because never an O is lost. O in and O out is basic.

Wrong.

>
>> No O is left in a place with a swap in and a swap out.
>> No O is left in a place with no swap in.
>>> Do you agree that in finite distance from start
>>> the 1:1 parity is guaranteed?
>> There is no finite distance in which
>> every exchange of X and O happens.
>
> For each place with an O out there is a place with an O in. Never an O gets lost. And that is what counts.

prove it.

>
> Regards, WM

Re: Natural numbers and vases

<t6u5so$96b$2@gioia.aioe.org>

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Subject: Re: Natural numbers and vases
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by: sergi o - Sat, 28 May 2022 21:57 UTC

On 5/28/2022 2:22 PM, WM wrote:
> Fritz Feldhase schrieb am Samstag, 28. Mai 2022 um 02:34:08 UTC+2:
>> On Friday, May 27, 2022 at 9:34:28 PM UTC+2, WM wrote:
>
>>> Nevertheless every exchange of X and O happens in finite distance from the start.
>> Yes, you are right.
>>
>> Hint: Jim Beam's chess examply is quite helpful for pointing out a fundamental error in your "argument". There are positions in chess which cannot be reached by any number of regular moves (starting with the initial position). Still this does not mean that these positions do not exist!
>
> But it means that they cannot be reached. Cantor claims that the complete indexing can be reached.

Cantor is correct.

>>
>> In the same way there *is* a matrix containing just the fractions 1/1, 2/1, 3/1, ...
>
> Of course, but it is much smaller than the matrix containing all fractions.

wrong, show why....

>
>> though it cannot be "reached" by any number of swaps.
>
> Whatever. Cantor claims that the complete indexing can be performed:

Wrong quote

>
> Not if we have to jump from an existing state to a not existing delusion.

you are in permanent delusion.

>
> Regards, WM

Re: Natural numbers and vases

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Sat, 28 May 2022 19:11:01 -0400
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by: Jim Burns - Sat, 28 May 2022 23:11 UTC

On 5/28/2022 3:52 PM, WM wrote:
> Jim Burns schrieb
> am Samstag, 28. Mai 2022 um 01:14:08 UTC+2:

>> This is possible because
>> there is no finite distance in which
>> every exchange of X and O happens.
>
> And there is no infinite distance and no completeness.

One kind of completeness is no last swap.
Another kind is each swap described.

We have the completeness of each swap described.
We have a description.

( Actually, by the second kind of completeness, the
( description, we know we _don't_ have the first
( kind of completeness, the end.

>> When all the swaps have swapped,
>> no O is left in a place with a swap out.
>> This is possible because
>> each place with a swap in has a swap out.
>>
>> When all swaps have been swapped,
>
> then there would be completeness.

We have the completeness of each swap described.

Starting with that, and proceeding only by
truth preserving inferences, we can arrive at
further claims which also have the completeness
of each swap described.

>> no O is left.
>
> That is a contradiction because never an O is lost.
> O in and O out is basic.

Infinity is not a reallyreallyreallyreallyreallyreally
large number. It is a different kind of thing.

We prove that here, by swapping the whole matrix
with its first column.

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, 29 May 2022 00:04 UTC

Re: Natural numbers and vases

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From: chris.m....@gmail.com (Chris M. Thomasson)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Sat, 28 May 2022 17:38:32 -0700
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by: Chris M. Thomasson - Sun, 29 May 2022 00:38 UTC

On 5/21/2022 1:34 PM, Jim Burns wrote:
> On 5/21/2022 3:38 PM, Chris M. Thomasson wrote:
>
>> I am not sure if the physical world is infinite.
>> Can we zoom in forever?
>> Also, can we zoom out forever?
>
> As far as zooming out is concerned...

Zooming out forever seems a bit plausible. Imagine if we are nothing
more than a speck within the infinite void? Sometimes I think we are
contained inside of a black hole residing in our "parent" universe...

Zooming out on the Mandelbrot set can make it look like a tiny speck,
only a single pixel, yet you can also zoom in on it forever...

>
> We have pretty firm observations for the age/size of
> the _observable_ universe.
> For that, the answer is "not forever".
>
> Suppose we very reasonably assume that, of all the
> universe that exists, observable or not, that part which
> we observe is nothing special.
>
> What we observe has a curvature very close to zero,
> with a 50/50 chance of being positive or negative.
>
> If the curvature is positive, the answer for the
> _whole_ universe is "not forever".
>
> If the curvature is negative,
> the _whole_ universe is literally infinite,
> and the answer is "forever".
>
>
> I think that some theorists are suspicious of
> how close the observed curvature is to zero
> (which is: reallyreallyreally damn close), and they're
> thinking about possible underlying principles that
> would force the curvature to be exactly zero.
> In that case, the answer would be "forever".
>
>
> As far as zooming in is concerned...
>
> There is a strong consensus that the answer
> is "not forever".
>
> The details are being worked out As We Speak,
> but everyone seems to agree that some further
> theory is needed to explain time, space, and gravity
> at the Very Small, and that, when we have it,
> time, space, and gravity won't look much like
> time, space, and gravity, so it is very likely
> that there is some zoom-in which won't make sense
> to call a "zoom-in". Thus "not forever".
>

Re: Natural numbers and vases

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From: chris.m....@gmail.com (Chris M. Thomasson)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Sat, 28 May 2022 17:38:58 -0700
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by: Chris M. Thomasson - Sun, 29 May 2022 00:38 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 - Sun, 29 May 2022 15:56 UTC

Jim Burns schrieb am Sonntag, 29. Mai 2022 um 01:11:12 UTC+2:
> On 5/28/2022 3:52 PM, WM wrote:
> > Jim Burns schrieb
> > am Samstag, 28. Mai 2022 um 01:14:08 UTC+2:
> >> This is possible because
> >> there is no finite distance in which
> >> every exchange of X and O happens.
> >
> > And there is no infinite distance and no completeness.
> One kind of completeness is no last swap.

That is not completeness. "Allow me to remark that the reality and the absolute principles of the integers appear to be much stronger than those of the world of sensations. And this fact has precisely one very simple reason, namely that the integers separately as well as in their actually infinite totality exist as eternal ideas in intellectu Divino in the highest degree of reality." [G. Cantor, letter to C. Hermite (30 Nov 1895)] Das God fail to think of the last part before omegain highest degree of reality?

> Another kind is each swap described.

And all swaps.
>
> We have the completeness of each swap described.
> We have a description.

Exchanging an X and an O from X.....O to O.....X will not make any of them disappear and will not make any of them multiply. It will simply maintain an X and an O. I would never have believed before that matheologians are dishonest enough (because not even an earthworm can be stupid enough) to contradict this simple 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 - Sun, 29 May 2022 15:58 UTC

Re: Natural numbers and vases

<97d27255-d796-4b1c-a30a-44226dba0c37n@googlegroups.com>

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https://www.novabbs.com/tech/article-flat.php?id=101447&group=sci.math#101447

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
Injection-Date: Sun, 29 May 2022 19:56:25 +0000
Content-Type: text/plain; charset="UTF-8"

by: William - Sun, 29 May 2022 19:56 UTC

Re: Natural numbers and vases

<7d9f82cc-e322-468d-8deb-bcdbc8e11e3cn@googlegroups.com>

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https://www.novabbs.com/tech/article-flat.php?id=101453&group=sci.math#101453

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Sun, 29 May 2022 20:36 UTC

Re: Natural numbers and vases

<eba05ea8-58c8-2b48-969c-9989d8e0c457@att.net>

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Sun, 29 May 2022 17:33:40 -0400
Organization: A noiseless patient Spider
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by: Jim Burns - Sun, 29 May 2022 21:33 UTC

On 5/29/2022 11:56 AM, WM wrote:
> Jim Burns schrieb
> am Sonntag, 29. Mai 2022 um 01:11:12 UTC+2:
>> On 5/28/2022 3:52 PM, WM wrote:
>>> Jim Burns schrieb
>>> am Samstag, 28. Mai 2022 um 01:14:08 UTC+2:

>>>> This is possible because
>>>> there is no finite distance in which
>>>> every exchange of X and O happens.
>>>
>>> And there is no infinite distance and
>>> no completeness.
>>
>> One kind of completeness is no last swap.
>
> That is not completeness.

s/completeness/incompleteness

>> Another kind is each swap described.
>
> And all swaps.
>
>> We have the completeness of each swap described.
>> We have a description.
>
> Exchanging an X and an O from X.....O to O.....X
> will not make any of them disappear and will not
> make any of them multiply.

They do not disappear. They do not multiply.
They match a proper subset.

Not all collections are unable to match a proper subset.
We prove that here.

> It will simply maintain an X and an O.
> I would never have believed before that
> matheologians are dishonest enough
> (because not even an earthworm can be stupid enough)
> to contradict this simple fact.

This simple fact?

1/1:1/1 1/2:2/1 1/3:4/1 1/4:7/1 ...
2/1:3/1 2/2:5/1 2/3:8/1 2/4:12/1 ...
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8/1:36/1 8/2:44/1 8/3:53/1 8/4:63/1 ...
9/1:45/1 9/2:54/1 9/3:64/1 9/4:75/1 ...
.... ... ... ...

Re: Natural numbers and vases

<t70qs2$a89$1@dont-email.me>

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From: nom...@afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Sun, 29 May 2022 18:07:55 -0700
Organization: Peripheral Visions
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by: FromTheRafters - Mon, 30 May 2022 01:07 UTC

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
Injection-Date: Sun, 29 May 2022 22:33:50 +0000
Content-Type: text/plain; charset="UTF-8"
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by: William - Sun, 29 May 2022 22:33 UTC

For large values of one, one equals two, for small values of two.

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

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.