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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=103206&group=sci.math#103206

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From: inva...@invalid.com (sergi o)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Tue, 14 Jun 2022 14:23:00 -0500
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by: sergi o - Tue, 14 Jun 2022 19:23 UTC

Re: Natural numbers and vases

<t8annl$nsb$1@gioia.aioe.org>

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From: inva...@invalid.com (sergi o)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Tue, 14 Jun 2022 14:32:05 -0500
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by: sergi o - Tue, 14 Jun 2022 19:32 UTC

Re: Natural numbers and vases

<87ilp2lp06.fsf@bsb.me.uk>

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From: ben.use...@bsb.me.uk (Ben Bacarisse)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Wed, 15 Jun 2022 02:00:25 +0100
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by: Ben Bacarisse - Wed, 15 Jun 2022 01:00 UTC

Fritz Feldhase <franz.fritschee.ff@gmail.com> writes:

> On Tuesday, June 14, 2022 at 4:23:54 AM UTC+2, Ben Bacarisse wrote:
>> sergi o <inv...@invalid.com> writes:
>>
>> > On 6/13/2022 3:01 PM, WM wrote:
>> >> Fritz Feldhase schrieb am Montag, 13. Juni 2022 um 20:01:47 UTC+2:
>> >>> On Monday, June 13, 2022 at 4:21:57 PM UTC+2, WM wrote:
>> >>>> Fritz Feldhase schrieb am Sonntag, 12. Juni 2022 um 19:20:14 UTC+2:
>> >>>>> On Sunday, June 12, 2022 at 6:58:30 PM UTC+2, WM wrote:
>> >>>>>
>> >>>>> Look, you psychotic asshole, your claim was: "Every extensionally given set is well-ordered by the order of writing."
>> >>>>>
>> >>>>> Now let M = {elephant, donkey, walrus} = {donkey, elephant, walrus} = {walrus, donkey, elephant} = ...
>> >>>>>
>> >>>>> Which *is* the smallest element in M?
>> >>>>>
>> >>>> Well-order has nothing to do with largeness.
>> >>>
>> >>> Oh, really?
>> >>
>> >> Really.
>> >>
>> > How can you do any ordering without size?
>> >
>> The relation R = {(elephant, donkey), (elephant, walrus), (donkey, walrus)}
>> is a strict (irreflexive) total order on M.
>
> Implying elephant < donkey < walrus (with < := R).
>
> Hence elephant is the least/smallest element in M (in respect to this order).
>
> Hint: "In mathematics, especially in order theory, the /least element/ of a subset S of a partially ordered set (poset) is an element of S that is smaller than every other element of S."
>
> You see: elephant is _smaller_ than donkey and walrus.

There are other ways to phrase it. We don't have to use the language of
size and, often, when we do, it gets in the way. You end up with
students puzzling about well-ordering R "by size".

> That's why Sergio's question is (at least) sensible and WM's claim is
> braindead nonsense

I didn't want to suggest his question was not sensible. I wanted to
show that you don't have to refer to size if you don't want to. You can
express ∃ x ∈ S, ∀ y ∈ S x=y or R(x,y) with no need to use any metaphor
for the relation.

And why is WM's statement braindead nonsense? (I can't imagine I typed
that sentence since almost everything he writes is nonsense, but in
this case it seems perfectly sane.)

> EOD

I didn't see this. Sorry.

--
Ben.

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 - Wed, 15 Jun 2022 05:17 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 - Thu, 16 Jun 2022 17:52 UTC

Re: Natural numbers and vases

<t8fsfm$cd7$1@dont-email.me>

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https://www.novabbs.com/tech/article-flat.php?id=103345&group=sci.math#103345

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From: nom...@afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Thu, 16 Jun 2022 14:23:45 -0400
Organization: Peripheral Visions
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by: FromTheRafters - Thu, 16 Jun 2022 18:23 UTC

Re: Natural numbers and vases

<32751fd2-b447-4c3d-87a5-1e9418691df6n@googlegroups.com>

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Thu, 16 Jun 2022 21:12 UTC

Re: Natural numbers and vases

<t8g88j$164a$1@gioia.aioe.org>

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https://www.novabbs.com/tech/article-flat.php?id=103368&group=sci.math#103368

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From: inva...@invalid.com (sergi o)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Thu, 16 Jun 2022 16:44:50 -0500
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by: sergi o - Thu, 16 Jun 2022 21:44 UTC

Re: Natural numbers and vases

<fdaef085-ae4f-47d6-8d7d-bc70cc3a1b99n@googlegroups.com>

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Subject: Re: Natural numbers and vases
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Fri, 17 Jun 2022 00:51 UTC

Re: Natural numbers and vases

<6d5663d5-1f15-4864-a454-5d17ec41b5dan@googlegroups.com>

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Subject: Re: Natural numbers and vases
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Fri, 17 Jun 2022 07:15 UTC

Re: Natural numbers and vases

<37ed58e0-de2c-4457-9692-e0ab1ba10e26n@googlegroups.com>

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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 - Fri, 17 Jun 2022 09:16 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 - Fri, 17 Jun 2022 10:44 UTC

On Thursday, June 16, 2022 at 7:53:06 PM UTC+2, WM wrote:
> zelos...@gmail.com schrieb am Mittwoch, 15. Juni 2022 um 07:17:36 UTC+2:
> >
> > [....] what you want to do is
> >
> > | ℕ \ {1} \ {1, 2} \ {1, 2, 3} \ ... |
> >
> > Which is indeed, 0
> >
> Not when only subtracting FISONs smaller than ℕ.

Look, dumbo, there are no other FISONs. Each and every FISON is "smaller than" IN.

Hint: Each and every FISON is FINITE, but IN is INFINITE. HENCE each and every FISON is "smaller than" IN.

Too complicated for your bird brain?

Now let's consider the claim:

| ℕ \ {1} \ {1, 2} \ {1, 2, 3} \ ... | = 0

=============== repost =============
Actually, this point is quite interesting (from a mathematical point of view) imho.

Set theory does not allow for infinitely many applications of a binary operation "at once" (or iterated).

Though we may have an intuitive idea of the meaning of, say,

{1} u {1, 2} u {1, 2, 3} u ...

we simply "can't do" that [i.e. applying infinitely many operations, iterated] in set theory. (Note that there are no infinite expressions in set theory.)

The trick in this case is the notion of the (set-theoretic) union of a set.

Hence we might consider

U{{1, ..., n} : n e IN}

instead.

Of course, NOW we may define _the expression_ "{1} u {1, 2} u {1, 2, 3} u ...." the following way:

{1} u {1, 2} u {1, 2, 3} u ... =df U{{1, ..., n} : n e IN}.

____________

Concerning the presssent topic, it seems to me that WM has something in mind like

IN \ F1 \ F2 \ F3 \ ... (*)

But this [applying infinitely many operations, iterated] (again) "cannot be done" in the context of set theory, though we might have an intuitive idea of "the meaning" of (*).

IF we are willing to agree that the "result" of (*) should be a set X such that X contains all elements that are in IN but in none of the (infinitely many) sets F1, F2, F3, ... THEN

IN \ U{Fi : Fi e IN}

seems to be an appropriate formalization of (*).

Actually, I would like to define the notion "set-theoretic difference" the following way:

A \\ X =d {x e A : x !e UX}

Now we may define, say, the expression "IN \ {1} \ {1, 2} \ {1, 2, 3} \ ..." the following way:

IN \ {1} \ {1, 2} \ {1, 2, 3} \ ... =df IN \\ {{1, ..., n} : n e IN}.

With the definition A \ B =df {x e A : x !e B} it's easy to prove than that, say,

(A \ B1) \ B2 = A \\ {B1, B2}.

And we may meta-prove that this equation holds for any (finite) number of terms Bi.

In any case, we may consider statements like

IN \ {1} \ {1, 2} \ {1, 2, 3} \ ... = { }

now.

================ end of repost =============
NOW:

Theorem: | IN \ {1} \ {1, 2} \ {1, 2, 3} \ ... | = 0.

Proof: IN \ {1} \ {1, 2} \ {1, 2, 3} \ ... = IN \\ {{1, ..., n} : n e IN} = IN \ U{{1, ..., n} : n e IN} = IN \ IN = { }. Hence | IN \ {1} \ {1, 2} \ {1, 2, 3} \ ... | = | { } | = 0. qed

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 - Fri, 17 Jun 2022 11:10 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 - Fri, 17 Jun 2022 12:53 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, 17 Jun 2022 12:58 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 - Fri, 17 Jun 2022 13:09 UTC

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: franz.fr...@gmail.com (Fritz Feldhase)
Injection-Date: Fri, 17 Jun 2022 13:14:28 +0000
Content-Type: text/plain; charset="UTF-8"

by: Fritz Feldhase - Fri, 17 Jun 2022 13:14 UTC

Re: Natural numbers and vases

<t8i0eb$d0g$1@gioia.aioe.org>

copy mid

https://www.novabbs.com/tech/article-flat.php?id=103429&group=sci.math#103429

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Path: i2pn2.org!i2pn.org!aioe.org!jq9Zon5wYWPEc6MdU7JpBw.user.46.165.242.75.POSTED!not-for-mail
From: inva...@invalid.com (sergi o)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Fri, 17 Jun 2022 08:43:38 -0500
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by: sergi o - Fri, 17 Jun 2022 13:43 UTC

Re: Natural numbers and vases

<t8i0hv$d0g$2@gioia.aioe.org>

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Subject: Re: Natural numbers and vases
Date: Fri, 17 Jun 2022 08:45:34 -0500
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by: sergi o - Fri, 17 Jun 2022 13:45 UTC

Re: Natural numbers and vases

<c9593ec9-828a-4f46-b3e2-189ee1529d18n@googlegroups.com>

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Subject: Re: Natural numbers and vases
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by: WM - Fri, 17 Jun 2022 14:16 UTC

Re: Natural numbers and vases

<4671d301-3fe3-4e74-83a8-e795d8fe89fen@googlegroups.com>

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Fri, 17 Jun 2022 14:18 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, 17 Jun 2022 14:53 UTC

Re: Natural numbers and vases

<t8i5rh$s09$1@gioia.aioe.org>

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https://www.novabbs.com/tech/article-flat.php?id=103441&group=sci.math#103441

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Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Fri, 17 Jun 2022 10:16:00 -0500
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by: sergi o - Fri, 17 Jun 2022 15:16 UTC

Re: Natural numbers and vases

<t8i5t8$s09$2@gioia.aioe.org>

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https://www.novabbs.com/tech/article-flat.php?id=103442&group=sci.math#103442

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From: inva...@invalid.com (sergi o)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Fri, 17 Jun 2022 10:16:55 -0500
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by: sergi o - Fri, 17 Jun 2022 15:16 UTC

Re: Natural numbers and vases

<t8m7qc$dba$1@gioia.aioe.org>

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https://www.novabbs.com/tech/article-flat.php?id=103598&group=sci.math#103598

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From: inva...@invalid.com (sergi o)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Sat, 18 Jun 2022 23:14:03 -0500
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by: sergi o - Sun, 19 Jun 2022 04:14 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.