novaBBS - 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=104833&group=sci.math#104833

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
Injection-Date: Wed, 29 Jun 2022 16:50:18 +0000
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by: WM - Wed, 29 Jun 2022 16:50 UTC

Re: Natural numbers and vases

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https://www.novabbs.com/tech/article-flat.php?id=104834&group=sci.math#104834

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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by: William - Wed, 29 Jun 2022 16:56 UTC

Re: Natural numbers and vases

<c92c9a16-1eac-4bd4-8429-06a4362b0683n@googlegroups.com>

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https://www.novabbs.com/tech/article-flat.php?id=104835&group=sci.math#104835

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Wed, 29 Jun 2022 16:56 UTC

Re: Natural numbers and vases

<e35cd52f-3a06-4d41-8707-295f849877dbn@googlegroups.com>

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https://www.novabbs.com/tech/article-flat.php?id=104836&group=sci.math#104836

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Wed, 29 Jun 2022 17:00 UTC

Re: Natural numbers and vases

<t9i0s9$hhm$1@gioia.aioe.org>

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Subject: Re: Natural numbers and vases
Date: Wed, 29 Jun 2022 12:07:19 -0500
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by: sergi o - Wed, 29 Jun 2022 17:07 UTC

Re: Natural numbers and vases

<7460d54b-c243-4516-957f-f02ecc4c25f5n@googlegroups.com>

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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by: William - Wed, 29 Jun 2022 17:09 UTC

Re: Natural numbers and vases

<6c25c484-1d21-4063-a910-614213f92f57n@googlegroups.com>

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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by: William - Wed, 29 Jun 2022 17:14 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 - Wed, 29 Jun 2022 17:14 UTC

Re: Natural numbers and vases

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https://www.novabbs.com/tech/article-flat.php?id=104842&group=sci.math#104842

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Wed, 29 Jun 2022 17:33 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 - Wed, 29 Jun 2022 17:42 UTC

Re: Natural numbers and vases

<5225b55a-9ac3-4596-be42-384068d6ce6dn@googlegroups.com>

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https://www.novabbs.com/tech/article-flat.php?id=104845&group=sci.math#104845

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Wed, 29 Jun 2022 17:45 UTC

Re: Natural numbers and vases

<ab9a3830-b7a4-4334-9794-3b6466137824n@googlegroups.com>

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https://www.novabbs.com/tech/article-flat.php?id=104848&group=sci.math#104848

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Wed, 29 Jun 2022 17:56 UTC

Re: Natural numbers and vases

<6e3a0e09-5d21-4774-b39e-6ad04d62fb67n@googlegroups.com>

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https://www.novabbs.com/tech/article-flat.php?id=104850&group=sci.math#104850

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
Injection-Date: Wed, 29 Jun 2022 17:59:41 +0000
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by: William - Wed, 29 Jun 2022 17:59 UTC

Re: Natural numbers and vases

<6dcf4354-6bf1-9b3b-ab7a-d9bd2140bdc9@att.net>

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https://www.novabbs.com/tech/article-flat.php?id=104851&group=sci.math#104851

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Wed, 29 Jun 2022 14:02:11 -0400
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by: Jim Burns - Wed, 29 Jun 2022 18:02 UTC

On 6/29/2022 12:48 PM, WM wrote:
> Fritz Feldhase schrieb
> am Mittwoch, 29. Juni 2022 um 18:39:43 UTC+2:
>> On Wednesday, June 29, 2022 at 6:18:33 PM UTC+2,
>> WM wrote:

>>> It follows by definition.
>>
>> You didn't state a proper definition
>
> The definition for ℕ_def has been given here:
> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀.

That is not a proper definition.

This is a proper definition:

<WM<JB<WM>>>
>>>
>>> By the axiom of separation these endsegments
>>> can be collected in a set ℕ_def such that
>>> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀ .
>>
>> The axiom of separation says that,
>> if there is a predicate
>> P(k) in which 'ℕ_def' does not occur,
>> then there is a set
>> ℕ_def = { k ∈ ℕ | P(k) }
>>
>> What is P(k) in which 'ℕ_def' does not occur?
>
> It is ℕ_def = { k ∈ ℕ | |E(k)| = ℵ₀ }
>
</WM<JB<WM>>>

Date: Thu, 23 Jun 2022 06:45:37 -0700 (PDT)
Subject: Re: Mapping exhausts, subtracting doesn't
Newsgroups: sci.logic
Message-ID: <0ea9ce6e-dec2-4114-a979-e70ddc1caafen@googlegroups.com>

You can't use 'ℕ_def' for telling us what 'ℕ_def' means.

It is a matter of effective communication,
when explaining/describing a term, to use
only terms which one's intended readers know.

One's intended readers for a definition of 'ℕ_def'
are those who _don't_ know what 'ℕ_def' means.

Using 'ℕ_def' to define 'ℕ_def' presents a
paradox-like situation. Only those who don't
need your definition can get anything of value
from your definition.
( Only paradox-like. There is a resolution.
( Don't do that.

> Only endsegments are collected which
> do not reduce the intersection below ℵ₀.

The same end segment in different collections
can "have" different intersections.
Membership in ℕ_def is not well-defined
as you state it here.

Of course,
_end segments_ don't "have" intersections.
_Collections_ of end segments have intersections.

The collection of end segments you want
seems to be of those for which
the end segments _up to_ have
an infinite end segment.

Collections _up to_ E(k) like these.
ENDS(k) = { E(j) | E(k) ⊆ E(j) }

with end segments like these
E(j) = { i ∈ ℕ | i =< j }

We can say E(k) does not
reduce the intersection below |ℕ|
like this
|⋂ENDS(k)| = |ℕ|

A better way to say what you want
seems to be
ℕ_def = { k ∈ ℕ : |⋂ENDS(k)| = |ℕ| }

If you instead define ℕ_def like that,
this is wrong
|∩{E(k) : k ∈ ℕ_def}| = ℵ₀.

It's still better, though,
because how you had it was not-even-wrong.

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 - Wed, 29 Jun 2022 20:00 UTC

Re: Natural numbers and vases

<2f40644b-ec85-42e1-bc80-b291fc2e82b9n@googlegroups.com>

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Wed, 29 Jun 2022 20:12 UTC

Jim Burns schrieb am Mittwoch, 29. Juni 2022 um 20:02:21 UTC+2:
> On 6/29/2022 12:48 PM, WM wrote:

> > The definition for ℕ_def has been given here:
> > |∩{E(k) : k ∈ ℕ_def}| = ℵ₀.
> That is not a proper definition.

It is. ℕ_def is the set of indices which make |∩{E(k) : k ∈ ℕ_def}| = ℵ₀.
>
> You can't use 'ℕ_def' for telling us what 'ℕ_def' means.

I dont. I use the intersection |∩{E(k) : k ∈ ℕ_def}| to tell you what ℕ_def means.
ℕ_def means all finite sets which yield an infinite Intersection of their endsegments.

|∩{E(1), E(2), ..., E(k)}| = ℵ₀
is true for k = 3, 4, 5, ...
But not for all natiral numbers.
>
> It is a matter of effective communication,
> when explaining/describing a term, to use
> only terms which one's intended readers know.
>
> One's intended readers for a definition of 'ℕ_def'
> are those who _don't_ know what 'ℕ_def' means.

Collect as many endsegments as you can which satisfy this equation
|∩{E(1), E(2), ..., E(k)}| = ℵ₀
and put them into the set {E(k) : k ∈ ℕ_def}.
Then try to solve the exercise |∩{E(k) : k ∈ ℕ_def}| = ?
Can you do it? Can you find the difference with
∩{E(k) : k ∈ ℕ} = { }?

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
Injection-Date: Wed, 29 Jun 2022 20:12:48 +0000
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by: William - Wed, 29 Jun 2022 20:12 UTC

On Wednesday, June 29, 2022 at 5:01:07 PM UTC-3, WM wrote:
> William schrieb am Mittwoch, 29. Juni 2022 um 19:59:52 UTC+2:
> > On Wednesday, June 29, 2022 at 2:45:58 PM UTC-3, WM wrote:
> >
> > > The Peano set has all last numbers but no last last number.
> > Piffle. A Perano set, like any set, does not change. Each element of a Peano set has a successor. Thus no element of the Peano set is a last element.
> Here you can learn the difference:
>
> This
> |∩{E(1), E(2), ..., E(k)}| = ℵ₀

true whenever there is a last k.

> is true for k = 3, 4, 5, ...
> Collect all endsegments which satisfy this equation in the set {E(k) : k ∈ ℕ_def}

a collection with a last k

> and take the intersection.

since the collection has a last k the intersection has cardinality ℵ₀

> Then try to solve the exercise |∩{E(k) : k ∈ ℕ_def}| = ?
> Can you do it? Can you find the difference with
> ∩{E(k) : k ∈ ℕ} = { }?

The difference is that N_def (which is not a Peano set) has a last element while |N (a Peano set) does not,

--
William Hughes

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: transfin...@gmail.com (Transfinity)
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by: Transfinity - Wed, 29 Jun 2022 20:20 UTC

William schrieb am Mittwoch, 29. Juni 2022 um 22:12:57 UTC+2:
> On Wednesday, June 29, 2022 at 5:01:07 PM UTC-3, WM wrote:
> > William schrieb am Mittwoch, 29. Juni 2022 um 19:59:52 UTC+2:
> > > On Wednesday, June 29, 2022 at 2:45:58 PM UTC-3, WM wrote:
> > >
> > > > The Peano set has all last numbers but no last last number.
> > > Piffle. A Perano set, like any set, does not change. Each element of a Peano set has a successor. Thus no element of the Peano set is a last element.
> > Here you can learn the difference:
> >
> > This
> > |∩{E(1), E(2), ..., E(k)}| = ℵ₀
> true whenever there is a last k.
> > is true for k = 3, 4, 5, ...
> > Collect all endsegments which satisfy this equation in the set {E(k) : k ∈ ℕ_def}
> a collection with a last k

But you cannot identify it, can you? A collection with a last k that cannot be identified is called a potentially infinite collection. You just accepted its existence.
>
> > and take the intersection.
>
> since the collection has a last k the intersection has cardinality ℵ₀

So it is . Since the last k cannot be found, it is potentially infinite, i.e., different from ℕ.

> > Then try to solve the exercise |∩{E(k) : k ∈ ℕ_def}| = ?
> > Can you do it? Can you find the difference with
> > ∩{E(k) : k ∈ ℕ} = { }?
> The difference is that N_def (which is not a Peano set) has a last element

Not a fixed one. Otherwise find it!

Regards, WM

Re: Natural numbers and vases

<t9ico4$r3$1@gioia.aioe.org>

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Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Wed, 29 Jun 2022 15:29:55 -0500
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by: sergi o - Wed, 29 Jun 2022 20:29 UTC

On 6/29/2022 3:20 PM, Transfinity wrote:
> William schrieb am Mittwoch, 29. Juni 2022 um 22:12:57 UTC+2:
>> On Wednesday, June 29, 2022 at 5:01:07 PM UTC-3, WM wrote:
>>> William schrieb am Mittwoch, 29. Juni 2022 um 19:59:52 UTC+2:
>>>> On Wednesday, June 29, 2022 at 2:45:58 PM UTC-3, WM wrote:
>>>>
>>>>> The Peano set has all last numbers but no last last number.
>>>> Piffle. A Perano set, like any set, does not change. Each element of a Peano set has a successor. Thus no element of the Peano set is a last element.
>>> Here you can learn the difference:
>>>
>>> This
>>> |∩{E(1), E(2), ..., E(k)}| = ℵ₀
>> true whenever there is a last k.
>>> is true for k = 3, 4, 5, ...
>>> Collect all endsegments which satisfy this equation in the set {E(k) : k ∈ ℕ_def}
>> a collection with a last k
>
> But you cannot identify it, can you? A collection with a last k that cannot be identified is called a potentially infinite collection.

no. When you stop at k, the intersection of E(1) to E(k) = E(k)
and as we all know |E(k)}| = ℵ₀

"identifying" it is red herring.

>>
>>> and take the intersection.
>>
>> since the collection has a last k the intersection has cardinality ℵ₀
>
>>> Then try to solve the exercise |∩{E(k) : k ∈ ℕ_def}| = ?
>>> Can you do it? Can you find the difference with
>>> ∩{E(k) : k ∈ ℕ} = { }?
>> The difference is that N_def (which is not a Peano set) has a last element
>
> Not a fixed one. Otherwise find it!
>

all sets are fixed.

> Regards, WM

Re: Natural numbers and vases

<c95bae54-1330-4273-98d3-b8c7742f3346n@googlegroups.com>

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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by: William - Wed, 29 Jun 2022 20:53 UTC

On Wednesday, June 29, 2022 at 5:20:25 PM UTC-3, Transfinity wrote:
> William schrieb am Mittwoch, 29. Juni 2022 um 22:12:57 UTC+2:
> > On Wednesday, June 29, 2022 at 5:01:07 PM UTC-3, WM wrote:
> > > William schrieb am Mittwoch, 29. Juni 2022 um 19:59:52 UTC+2:
> > > > On Wednesday, June 29, 2022 at 2:45:58 PM UTC-3, WM wrote:
> > > >
> > > > > The Peano set has all last numbers but no last last number.
> > > > Piffle. A Perano set, like any set, does not change. Each element of a Peano set has a successor. Thus no element of the Peano set is a last element.
> > > Here you can learn the difference:
> > >
> > > This
> > > |∩{E(1), E(2), ..., E(k)}| = ℵ₀
> > true whenever there is a last k.
> > > is true for k = 3, 4, 5, ...
> > > Collect all endsegments which satisfy this equation in the set {E(k) : k ∈ ℕ_def}
> > a collection with a last k
> But you cannot identify it, can you?

Nope, but I can prove that it exists That is all that is needed.
Note N_def is a set so it does not change. It is a bounded set so it has a largest element.

--
William Hughes

Re: Natural numbers and vases

<t9ief9$1hqkf$1@dont-email.me>

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Subject: Re: Natural numbers and vases
Date: Wed, 29 Jun 2022 16:59:16 -0400
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by: FromTheRafters - Wed, 29 Jun 2022 20:59 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 - Wed, 29 Jun 2022 22:06 UTC

William schrieb am Mittwoch, 29. Juni 2022 um 22:53:22 UTC+2:
> On Wednesday, June 29, 2022 at 5:20:25 PM UTC-3, Transfinity wrote:

> > > > > > The Peano set has all last numbers but no last last number.
> > > > > Piffle. A Perano set, like any set, does not change. Each element of a Peano set has a successor. Thus no element of the Peano set is a last element.
> > > > Here you can learn the difference:
> > > >
> > > > This
> > > > |∩{E(1), E(2), ..., E(k)}| = ℵ₀
> > > true whenever there is a last k.
> > > > is true for k = 3, 4, 5, ...
> > > > Collect all endsegments which satisfy this equation in the set {E(k) : k ∈ ℕ_def}
> > > a collection with a last k
> > But you cannot identify it, can you?
> Nope, but I can prove that it exists That is all that is needed.

You cannot. Whatever you prove, I can show that it is not the last endsegment of that set.

> Note N_def is a set so it does not change. It is a bounded set so it has a largest element.

Your note is as wrong as it could be. If ℕ_def does not change, then there is a fixed set with |∩{E(1), E(2), ..., E(k)}| = ℵ₀ . What happens if another endsegment is added? Will the intersection become finite or zero?

Regards, WM

Re: Natural numbers and vases

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by: WM - Wed, 29 Jun 2022 22:07 UTC

Re: Natural numbers and vases

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by: sergi o - Wed, 29 Jun 2022 23:06 UTC

On 6/29/2022 5:06 PM, WM wrote:
> William schrieb am Mittwoch, 29. Juni 2022 um 22:53:22 UTC+2:
>> On Wednesday, June 29, 2022 at 5:20:25 PM UTC-3, Transfinity wrote:
>
>>>>>>> The Peano set has all last numbers but no last last number.
>>>>>> Piffle. A Perano set, like any set, does not change. Each element of a Peano set has a successor. Thus no element of the Peano set is a last element.
>>>>> Here you can learn the difference:
>>>>>
>>>>> This
>>>>> |∩{E(1), E(2), ..., E(k)}| = ℵ₀
>>>> true whenever there is a last k.
>>>>> is true for k = 3, 4, 5, ...
>>>>> Collect all endsegments which satisfy this equation in the set {E(k) : k ∈ ℕ_def}
>>>> a collection with a last k
>>> But you cannot identify it, can you?
>> Nope, but I can prove that it exists That is all that is needed.
>
> You cannot. Whatever you prove, I can show that it is not the last endsegment of that set.
>
>> Note N_def is a set so it does not change. It is a bounded set so it has a largest element.
>
> Your note is as wrong as it could be. If ℕ_def does not change, then there is a fixed set with |∩{E(1), E(2), ..., E(k)}| = ℵ₀ . What happens if another endsegment is added? Will the intersection become finite or zero?
>

the only math you have there is this;

|∩{E(1), E(2), ..., E(k)}| = ℵ₀

which is true as ∩{E(1), E(2), ..., E(k)} = E(k)

and |{E(k)}| = ℵ₀

If you add another endsegment to the intersection, you have this;

|∩{E(1), E(2), ..., E(k+1)}| = ℵ₀

which is true as ∩{E(1), E(2), ..., E(k+1)} = E(k+1)

and |{E(k+1)}| = ℵ₀

the intersection obviously remains infinite.

Quit being lazy and do the equations!

However;

∩{E(1), E(2), ...} = {}

as the intersection of all is not an endsegment (for the 1,001th time you are told this)

> Regards, WM

Re: Natural numbers and vases

<18a58686-098a-4d1b-957e-47d162f66c32n@googlegroups.com>

copy mid

https://www.novabbs.com/tech/article-flat.php?id=104920&group=sci.math#104920

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
Injection-Date: Wed, 29 Jun 2022 23:25:17 +0000
Content-Type: text/plain; charset="UTF-8"

by: William - Wed, 29 Jun 2022 23:25 UTC

Matter will be damaged in direct proportion to its value.

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.