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Topic: Another AC anomaly?
Replies: 280   Last Post: Dec 27, 2009 3:53 PM

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Dik T. Winter

Posts: 7,899
Registered: 12/6/04
Re: Another AC anomaly?
Posted: Dec 8, 2009 10:13 AM
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In article <cccaffe3-72fe-4a1f-9514-a018e9379309@j24g2000yqa.googlegroups.com> WM <mueckenh@rz.fh-augsburg.de> writes:
> On 7 Dez., 16:37, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > In article <yvCdnW28VrXBqIXWnZ2dnUVZ_s-dn...@giganews.com> "K_h" <KHol...=
> @SX729.com> writes:
> > ...
> > > > > > When you mean with your statement about N:
> > > > > > N = union{n is natural} {n}
> > > > > > then that is not a limit. Check the definitions about
> > > > > > it.

> > > > >
> > > > > It is a limit. That is independent from any definition.

> > > >
> > > > It is not a limit. Nowhere in the definition of that
> > > > union a limit is used
> > > > or mentioned.

> > >
> > > Question. Isn't this simply a question of language?

> >
> > Not at all. When you define N as an infinite union there is no limit
> > involved, there is even no sequence involved. N follows immediately
> > from the axioms.

>
> Nevertheless it is a limit ordinal.


Yes, that does not mean that necessarily a limit is involved. It is a limit
in the sense that you do not get there by continuously getting at the
successor, in that case it is a limiting process. But when you define
N as an infinite union you do *not* go there by continuously getting at
a successor. The union of a collection (finite, countably infinite or
some other infinity) is defined whithout resorting to successor operations.
Moreover, they would even not make sens if the collection is infinite but
not countably infinite.

> > > Now w is a limit ordinal so the ordered set N is, in the
> > > ordinal sense, a limit. Of course w is not a member of N
> > > becasuse then N would be a member of itself (not allowed by
> > > foundation).

> >
> > Note here that N (the set of natural numbers) is *not* defined using a
> > limit at all. That w is called a limit ordinal is a definition of the
> > term "limit ordinal". It does not mean that the definition you use to
> > define it actually uses a limit. (And if I remember right, a limit
> > ordinal is an ordinal that has no predecessor, see, again no limit
> > involved.)

>
> Karel Hrbacek and Thomas Jech: "Introduction to Set Theory"
> Marcel Dekker Inc., New York, 1984, 2nd edition, p. 134
>
> As a starting point, we use the fact hat each natural number is
> identified with the set of all smaller natural numbers: n = {m in N :
> m < n}.


Note that here N is apparently already defined, without using a limit.

> Thus we let w, the least transfinite number, to be the set N
> of all natural numbers: w = N = {0, 1, 2, 3, ...}.
> It is easy to continue the process after this 'limit' step is made:
> The operation of successor can be used to produce numbers following w
> in the same way we used it to produce numbers following 0.


Yes. So what? That you can define things using a limit does *not*
imply that it is necessarily defined as a limit.
--
dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/


Date Subject Author
11/23/09
Read Re: Another AC anomaly?
Jesse F. Hughes
11/23/09
Read Re: Another AC anomaly?
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11/23/09
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11/24/09
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12/1/09
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11/30/09
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12/1/09
Read Jesse Hughes of Bentley College uses signature block to defame
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12/1/09
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12/1/09
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12/1/09
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others; Ann Reeves Re: What's wrong with supertasks?
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12/1/09
Read Re: using signature-blocks to defame innocent people Re: Jesse Hughes
of Bentley College
Marshall
12/2/09
Read Re: using signature-blocks to defame innocent people Re: Jesse Hughes
of Bentley College
Marshall
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/26/09
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mueckenh@rz.fh-augsburg.de
11/28/09
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mueckenh@rz.fh-augsburg.de
11/28/09
Read Re: Another AC anomaly?
Virgil
11/26/09
Read Re: Another AC anomaly?
William Hughes
11/28/09
Read Re: Another AC anomaly?
William Hughes
11/26/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/28/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
11/28/09
Read Re: Another AC anomaly?
Virgil
11/29/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/29/09
Read Re: Another AC anomaly?
Virgil
11/29/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
11/29/09
Read Re: Another AC anomaly?
Marshall
11/30/09
Read Re: Another AC anomaly?
Virgil
11/30/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
11/30/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
11/30/09
Read Re: Another AC anomaly?
Virgil
11/30/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
11/30/09
Read Re: Another AC anomaly?
Virgil
11/30/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
11/30/09
Read Re: Another AC anomaly?
Virgil
11/29/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/26/09
Read Re: Another AC anomaly?
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11/26/09
Read Re: Another AC anomaly?
William Hughes
11/26/09
Read Re: Another AC anomaly?
anonymous.rubbertube@yahoo.com
11/29/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
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11/27/09
Read Re: Another AC anomaly?
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11/29/09
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William Hughes
11/30/09
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mueckenh@rz.fh-augsburg.de
11/30/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
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11/27/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
Dik T. Winter
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/30/09
Read Re: Another AC anomaly?
Dik T. Winter
11/30/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/30/09
Read Re: Another AC anomaly?
Virgil
12/1/09
Read Re: Another AC anomaly?
Dik T. Winter
12/1/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/1/09
Read Re: Another AC anomaly?
Dik T. Winter
12/1/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/1/09
Read Re: Another AC anomaly?
William Hughes
12/1/09
Read Re: Another AC anomaly?
Virgil
12/1/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/1/09
Read Re: Another AC anomaly?
Virgil
12/1/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
12/1/09
Read Re: Another AC anomaly?
Virgil
12/1/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
12/1/09
Read Re: Another AC anomaly?
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12/2/09
Read Re: Another AC anomaly?
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12/2/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/2/09
Read Re: Another AC anomaly?
Dik T. Winter
12/2/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/2/09
Read Re: Another AC anomaly?
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12/3/09
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12/3/09
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mueckenh@rz.fh-augsburg.de
12/3/09
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Dik T. Winter
12/3/09
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12/7/09
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12/7/09
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12/8/09
Read Re: Another AC anomaly?
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12/8/09
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12/9/09
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12/9/09
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12/9/09
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12/9/09
Read Re: Another AC anomaly?
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12/9/09
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12/10/09
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12/9/09
Read Re: Another AC anomaly?
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12/9/09
Read Re: Another AC anomaly?
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12/10/09
Read Re: Another AC anomaly?
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12/10/09
Read Re: Another AC anomaly?
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12/10/09
Read Re: Another AC anomaly?
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12/10/09
Read Re: Another AC anomaly?
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12/11/09
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Read Re: Another AC anomaly?
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12/15/09
Read Re: Another AC anomaly?
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12/15/09
Read Re: Another AC anomaly?
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12/15/09
Read Re: Another AC anomaly?
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12/15/09
Read Re: Another AC anomaly?
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12/16/09
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12/16/09
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12/17/09
Read Re: Another AC anomaly?
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12/17/09
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12/17/09
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12/17/09
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12/17/09
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12/17/09
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12/18/09
Read Re: Another AC anomaly?
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12/18/09
Read Re: Another AC anomaly?
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12/18/09
Read Re: Another AC anomaly?
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12/19/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/19/09
Read Re: Another AC anomaly?
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12/22/09
Read Re: Another AC anomaly?
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12/22/09
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Virgil
12/18/09
Read Re: Another AC anomaly?
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12/19/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/19/09
Read Re: Another AC anomaly?
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12/21/09
Read Re: Another AC anomaly?
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12/21/09
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mueckenh@rz.fh-augsburg.de
12/21/09
Read Re: Another AC anomaly?
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12/21/09
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12/22/09
Read Re: Another AC anomaly?
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12/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/27/09
Read Re: Another AC anomaly?
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12/16/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/16/09
Read Re: Another AC anomaly?
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12/11/09
Read Re: Another AC anomaly?
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12/11/09
Read Re: Another AC anomaly?
Dik T. Winter
12/11/09
Read Re: Another AC anomaly?
K_h
12/11/09
Read Re: Another AC anomaly?
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12/12/09
Read Re: Another AC anomaly?
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12/12/09
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12/12/09
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12/11/09
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mueckenh@rz.fh-augsburg.de
12/11/09
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12/10/09
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mueckenh@rz.fh-augsburg.de
12/10/09
Read Re: Another AC anomaly?
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12/10/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/10/09
Read Re: Another AC anomaly?
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12/10/09
Read Re: Another AC anomaly?
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12/10/09
Read Re: Another AC anomaly?
Dik T. Winter
12/11/09
Read Re: Another AC anomaly?
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12/11/09
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12/7/09
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mueckenh@rz.fh-augsburg.de
12/7/09
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12/8/09
Read Re: Another AC anomaly?
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12/8/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/8/09
Read Re: Another AC anomaly?
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12/8/09
Read Re: Another AC anomaly?
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12/8/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/8/09
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Virgil
12/10/09
Read Re: Another AC anomaly?
Dik T. Winter
12/3/09
Read Re: Another AC anomaly?
Virgil
12/3/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/3/09
Read Re: Another AC anomaly?
Virgil
12/7/09
Read Re: Another AC anomaly?
Dik T. Winter
12/7/09
Read Re: Another AC anomaly?
Virgil
12/7/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/7/09
Read Re: Another AC anomaly?
Virgil
12/8/09
Read Re: Another AC anomaly?
Dik T. Winter
12/8/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/8/09
Read Re: Another AC anomaly?
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12/10/09
Read Re: Another AC anomaly?
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12/10/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/10/09
Read Re: Another AC anomaly?
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12/10/09
Read Re: Another AC anomaly?
Dik T. Winter
12/11/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/11/09
Read Re: Another AC anomaly?
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12/11/09
Read Re: Another AC anomaly?
Virgil
12/15/09
Read Re: Another AC anomaly?
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12/16/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/16/09
Read Re: Another AC anomaly?
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12/16/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/17/09
Read Re: Another AC anomaly?
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12/17/09
Read Re: Another AC anomaly?
Dik T. Winter
12/4/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/4/09
Read Re: Another AC anomaly?
Virgil
12/4/09
Read Re: Another AC anomaly?
Marshall
12/7/09
Read Re: Another AC anomaly?
Dik T. Winter
12/7/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/7/09
Read Re: Another AC anomaly?
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12/8/09
Read Re: Another AC anomaly?
Dik T. Winter
12/8/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/8/09
Read Re: Another AC anomaly?
Virgil
12/10/09
Read Re: Another AC anomaly?
Dik T. Winter
12/10/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/10/09
Read Re: Another AC anomaly?
Virgil
12/10/09
Read Re: Another AC anomaly?
Dik T. Winter
12/11/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/11/09
Read Re: Another AC anomaly?
Marshall
12/11/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/11/09
Read Re: Another AC anomaly?
Virgil
12/11/09
Read Re: Another AC anomaly?
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12/11/09
Read Re: Another AC anomaly?
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12/12/09
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12/12/09
Read Re: Another AC anomaly?
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12/12/09
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12/12/09
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12/12/09
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12/12/09
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12/12/09
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12/12/09
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12/13/09
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12/15/09
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12/16/09
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12/17/09
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12/17/09
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12/17/09
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12/18/09
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12/18/09
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12/18/09
Read Re: Another AC anomaly?
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12/19/09
Read Re: Another AC anomaly?
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12/19/09
Read Re: Another AC anomaly?
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12/19/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/19/09
Read Re: Another AC anomaly?
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12/19/09
Read Re: Another AC anomaly?
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12/19/09
Read Re: Another AC anomaly?
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12/19/09
Read Re: Another AC anomaly?
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12/19/09
Read Re: Another AC anomaly?
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12/22/09
Read Re: Another AC anomaly?
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Read Re: Another AC anomaly?
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12/27/09
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12/21/09
Read Re: Another AC anomaly?
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12/21/09
Read Re: Another AC anomaly?
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12/21/09
Read Re: Another AC anomaly?
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12/22/09
Read Re: Another AC anomaly?
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12/27/09
Read Re: Another AC anomaly?
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12/27/09
Read Re: Another AC anomaly?
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12/27/09
Read Re: Another AC anomaly?
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12/27/09
Read Re: Another AC anomaly?
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12/27/09
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12/27/09
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12/13/09
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12/4/09
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12/4/09
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12/4/09
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12/4/09
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12/4/09
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12/5/09
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12/5/09
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12/5/09
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12/6/09
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12/5/09
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12/2/09
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Virgil
12/1/09
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12/1/09
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11/27/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
anonymous.rubbertube@yahoo.com
11/30/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
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11/27/09
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11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
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11/27/09
Read Re: Another AC anomaly?
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11/27/09
Read Re: Another AC anomaly?
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11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
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11/27/09
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11/27/09
Read Re: Another AC anomaly?
Virgil

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