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

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mueckenh@rz.fh-augsburg.de

Posts: 18,076
Registered: 1/29/05
Re: Another AC anomaly?
Posted: Dec 13, 2009 3:47 PM
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On 13 Dez., 20:49, Virgil <Vir...@home.esc> wrote:

> > > Yes. It was riddled with obvious errors.
>
> > Which error was obvious?
>
> Among many others, your claim that in a complete infinite binary tree
> there are as many nodes as sets of nodes.


So it may seem to you. Your impression is wrong.

The binary tree contains all real numbers between 0 and 1. Each one is
represented by one single path, some are represented by two paths. If
every node is covered, then every real number of [0, 1] has been
constructed.

Remember the ZF-law for unions of index sets:
{1} U {1, 2} U {1, 2, 3} U ... = {1, 2, 3, ...}
This means: After having constructed all the paths ending with tails
of zeros

0.1000...
0.11000...
0.111000...
...
also the path
0.111...
has been constructed.
(The indexes of the 1' in above paths are the finite sets {1}, {1, 2},
{1, 2, 3}, ...)

Then you might think you could choose as an alternative "tail"
010101...
As you already constructed in the first run all paths of the form

0.000...
0.01000...
0.0001000...
...
you have also constructed the path
0.010101010...
because the indexes of the finite bit sequences in above paths are the
finite sets {1}, {1, 2}, {1, 2, 3}, ... . And the infinite set {1, 2,
3, ...} of them includes already the bit sequence 0.010101 of the path
1/3.

We have, if ZF is correct,
{1} U {1, 2} U {1, 2, 3} U ... = {1, 2, 3, ...}
That means, you have constructed with the first approach using tail
000... already all paths that are possible in the tree.
You could have started also with any other choice of the "tail", say
that of 1/pi - with the same result.

Therefore, the set of paths has been constructed by countably many
steps. Once in a while, two paths have been constructed
simultaneously, namely when an infinite sequence has been "finished".
Then you have obtained 1 + aleph_0 paths instead of only aleph_0
paths. But is there a difference?

The final result is: aleph_0 paths have been used to construct the
complete infinite binary tree with all its paths such that no further
path remains to be constructed, because the tree has been exhausted.

Regards, WM




Date Subject Author
11/23/09
Read Re: Another AC anomaly?
Jesse F. Hughes
11/23/09
Read Re: Another AC anomaly?
Herman Jurjus
11/23/09
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master1729
11/25/09
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T.H. Ray
11/24/09
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george
12/1/09
Read Re: Another AC anomaly?
george
11/25/09
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Bill Taylor
11/26/09
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Daryl McCullough
11/30/09
Read What's wrong with supertasks? (was Re: Another AC anomaly?)
Herman Jurjus
12/1/09
Read Jesse Hughes of Bentley College uses signature block to defame
others; Ann Reeves Re: What's wrong with supertasks?
plutonium.archimedes@gmail.com
12/1/09
Read Re: Jesse Hughes of Bentley College uses signature block to defame
others; Ann Reeves Re: What's wrong with supertasks?
Marshall
12/1/09
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Bentley College
plutonium.archimedes@gmail.com
12/1/09
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others; Ann Reeves Re: What's wrong with supertasks?
Seth Breidbart
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
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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
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William Hughes
11/28/09
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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ross.finlayson@gmail.com
11/28/09
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Virgil
11/29/09
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mueckenh@rz.fh-augsburg.de
11/29/09
Read Re: Another AC anomaly?
Virgil
11/29/09
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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
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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
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ross.finlayson@gmail.com
11/30/09
Read Re: Another AC anomaly?
Virgil
11/30/09
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ross.finlayson@gmail.com
11/30/09
Read Re: Another AC anomaly?
Virgil
11/29/09
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mueckenh@rz.fh-augsburg.de
11/26/09
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LauLuna
11/26/09
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William Hughes
11/26/09
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anonymous.rubbertube@yahoo.com
11/29/09
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William Hughes
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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Virgil
11/27/09
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Alan Smaill
11/29/09
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William Hughes
11/30/09
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mueckenh@rz.fh-augsburg.de
11/30/09
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Virgil
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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Virgil
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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Virgil
11/27/09
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William Hughes
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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Alan Smaill
11/27/09
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Virgil
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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Dik T. Winter
11/27/09
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mueckenh@rz.fh-augsburg.de
11/30/09
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Dik T. Winter
11/30/09
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mueckenh@rz.fh-augsburg.de
11/30/09
Read Re: Another AC anomaly?
Virgil
12/1/09
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Dik T. Winter
12/1/09
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mueckenh@rz.fh-augsburg.de
12/1/09
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Dik T. Winter
12/1/09
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mueckenh@rz.fh-augsburg.de
12/1/09
Read Re: Another AC anomaly?
William Hughes
12/1/09
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Virgil
12/1/09
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mueckenh@rz.fh-augsburg.de
12/1/09
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Virgil
12/1/09
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ross.finlayson@gmail.com
12/1/09
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Virgil
12/1/09
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ross.finlayson@gmail.com
12/1/09
Read Re: Another AC anomaly?
Virgil
12/2/09
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Dik T. Winter
12/2/09
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mueckenh@rz.fh-augsburg.de
12/2/09
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Dik T. Winter
12/2/09
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mueckenh@rz.fh-augsburg.de
12/2/09
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12/3/09
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Dik T. Winter
12/3/09
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mueckenh@rz.fh-augsburg.de
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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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12/10/09
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12/10/09
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12/10/09
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mueckenh@rz.fh-augsburg.de
12/10/09
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12/10/09
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12/11/09
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12/15/09
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12/15/09
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12/15/09
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12/15/09
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12/16/09
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12/16/09
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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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12/17/09
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12/18/09
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mueckenh@rz.fh-augsburg.de
12/18/09
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12/18/09
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12/19/09
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mueckenh@rz.fh-augsburg.de
12/19/09
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12/22/09
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Dik T. Winter
12/22/09
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Virgil
12/18/09
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12/19/09
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12/19/09
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12/21/09
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12/21/09
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mueckenh@rz.fh-augsburg.de
12/21/09
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12/21/09
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12/22/09
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Dik T. Winter
12/27/09
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mueckenh@rz.fh-augsburg.de
12/27/09
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12/16/09
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mueckenh@rz.fh-augsburg.de
12/16/09
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Virgil
12/11/09
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K_h
12/11/09
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12/11/09
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K_h
12/11/09
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Marshall
12/12/09
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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
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Virgil
12/10/09
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mueckenh@rz.fh-augsburg.de
12/10/09
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12/10/09
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12/10/09
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Dik T. Winter
12/11/09
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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
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12/8/09
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12/8/09
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anonymous.rubbertube@yahoo.com
12/8/09
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12/8/09
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mueckenh@rz.fh-augsburg.de
12/8/09
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Virgil
12/10/09
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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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12/7/09
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Dik T. Winter
12/7/09
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12/7/09
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mueckenh@rz.fh-augsburg.de
12/7/09
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Virgil
12/8/09
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Dik T. Winter
12/8/09
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mueckenh@rz.fh-augsburg.de
12/8/09
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12/10/09
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Dik T. Winter
12/10/09
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mueckenh@rz.fh-augsburg.de
12/10/09
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12/10/09
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Dik T. Winter
12/11/09
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mueckenh@rz.fh-augsburg.de
12/11/09
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12/11/09
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Virgil
12/15/09
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Dik T. Winter
12/16/09
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mueckenh@rz.fh-augsburg.de
12/16/09
Read Re: Another AC anomaly?
Dik T. Winter
12/16/09
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mueckenh@rz.fh-augsburg.de
12/17/09
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T.H. Ray
12/17/09
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Dik T. Winter
12/4/09
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mueckenh@rz.fh-augsburg.de
12/4/09
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Virgil
12/4/09
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Marshall
12/7/09
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Dik T. Winter
12/7/09
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mueckenh@rz.fh-augsburg.de
12/7/09
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Virgil
12/8/09
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Dik T. Winter
12/8/09
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mueckenh@rz.fh-augsburg.de
12/8/09
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Virgil
12/10/09
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Dik T. Winter
12/10/09
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mueckenh@rz.fh-augsburg.de
12/10/09
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Virgil
12/10/09
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Dik T. Winter
12/11/09
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mueckenh@rz.fh-augsburg.de
12/11/09
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Marshall
12/11/09
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mueckenh@rz.fh-augsburg.de
12/11/09
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Virgil
12/11/09
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Virgil
12/11/09
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Marshall
12/12/09
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mueckenh@rz.fh-augsburg.de
12/12/09
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Virgil
12/12/09
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Marshall
12/12/09
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mueckenh@rz.fh-augsburg.de
12/12/09
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Virgil
12/12/09
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Marshall
12/12/09
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mueckenh@rz.fh-augsburg.de
12/12/09
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Virgil
12/12/09
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Marshall
12/12/09
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12/12/09
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12/12/09
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george
12/13/09
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mueckenh@rz.fh-augsburg.de
12/13/09
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12/15/09
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12/16/09
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mueckenh@rz.fh-augsburg.de
12/16/09
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12/17/09
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12/17/09
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mueckenh@rz.fh-augsburg.de
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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mueckenh@rz.fh-augsburg.de
12/18/09
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12/18/09
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Dik T. Winter
12/19/09
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mueckenh@rz.fh-augsburg.de
12/19/09
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Virgil
12/19/09
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mueckenh@rz.fh-augsburg.de
12/19/09
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Marshall
12/19/09
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Virgil
12/19/09
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Virgil
12/19/09
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ross.finlayson@gmail.com
12/19/09
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Virgil
12/22/09
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ross.finlayson@gmail.com
12/22/09
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12/27/09
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ross.finlayson@gmail.com
12/21/09
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12/21/09
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mueckenh@rz.fh-augsburg.de
12/21/09
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12/22/09
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Dik T. Winter
12/27/09
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mueckenh@rz.fh-augsburg.de
12/27/09
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12/27/09
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Virgil
12/27/09
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Virgil
12/27/09
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12/27/09
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12/13/09
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mueckenh@rz.fh-augsburg.de
12/13/09
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Virgil
12/4/09
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K_h
12/4/09
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mueckenh@rz.fh-augsburg.de
12/4/09
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Virgil
12/4/09
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mueckenh@rz.fh-augsburg.de
12/4/09
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Virgil
12/5/09
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mueckenh@rz.fh-augsburg.de
12/5/09
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Virgil
12/5/09
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mueckenh@rz.fh-augsburg.de
12/6/09
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Virgil
12/5/09
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Carsten Schultz
12/2/09
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Virgil
12/1/09
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george
12/1/09
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Virgil
11/27/09
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Virgil
11/27/09
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anonymous.rubbertube@yahoo.com
11/30/09
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William Hughes
11/27/09
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William Hughes
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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William Hughes
11/27/09
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anonymous.rubbertube@yahoo.com
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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Virgil
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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Virgil
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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William Hughes
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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Virgil

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