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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 12, 2009 12:02 PM
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On 12 Dez., 17:19, Marshall <marshall.spi...@gmail.com> wrote:
> On Dec 12, 1:32 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>

> > On 12 Dez., 02:01, Marshall <marshall.spi...@gmail.com> wrote:
>
> > > Showing
> > > a contradiction would qualify, but it's been well
> > > established that you don't know how to do that.

>
> > Consider how a union of paths is counted (I copy
> > from another posting, therefore the quotation symbols):

>
> Ascii diagrams don't qualify as a contradiction.


Why should pictures, diagrams, acoustic signals etc. qualify less than
sequences of symbols? Every bit of information, in what form ever, can
be used in proofs. But here you are:

The union of all natural numbers is, according to set theory, omega.
If *actual* infinity is meant, this is plainly impossible, because the
natural numbers count themselves. So every set of natural numbers has
at most one number that is as large as its cardinal number. Using
initial sets of only even numbers, this becomes immediately clear:
2
2, 4
2, 4, 6
...
These sets contain always numbers that are larger than the cardinal
number of the set.
But this fact does not stop set theorists from believing in actual
infinity. They do not wish to see that ordinal number = cardinal
number as long as only initial segments of natural numbers are
involved.

Therefore I have looked for a model which supplies ordinal and
cardinal numbers in one and the same elements. This model is the
binary tree. If you union paths, then you get the sequence


> {1}
> {1, 2}
> {1, 2, 3}
> ...
> {1, 2, 3, ...}
> {1, 2, 3, ...}



i.e., when you union only all finite initial segments of 0.000... you
get exactly the same result as when you union all finite initial
segments of 0.000... *and* 0.000...

This leads to the result that the same structure, namely the tree with
all its nodes, contains only a countable set of paths and
simultaneously it contains an uncountable set of paths.

And this is a contradiction.

Of course, if only a small collection of paths from the tree is
presented, then one can find paths in the tree which are not included
in that collection.

But this does not help to explain an uncountable set of paths:
1) We can construct the tree from its finite paths. In order to obtain
one infinite "bonus" path, you need aleph_0 finite paths. But in order
to explain the existence of uncountably many infinite paths, you would
need to obtain at least aleph_0 infinite paths with every constructed
finite path - and even that would not be enough because aleph_0 *
aleph_0 = aleph_0.
2) If you construct the tree by means of infinite paths, then the
ratio between constructed paths and constructed nodes is 1/aleph_0 at
every step of the construction for every set of nodes covered by the
construction. Do you seriously expect that this ratio can become
2^aleph_0 - after all nodes will have been constructed (not *during*
construction, because as long as one node has not been constructed,
the ratio remains the same)?

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
Read Re: Another AC anomaly?
master1729
11/25/09
Read Re: Another AC anomaly?
T.H. Ray
11/24/09
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12/1/09
Read Re: Another AC anomaly?
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11/25/09
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11/26/09
Read Re: Another AC anomaly?
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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
Read using signature-blocks to defame innocent people Re: Jesse Hughes of
Bentley College
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?
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
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
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mueckenh@rz.fh-augsburg.de
11/28/09
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ross.finlayson@gmail.com
11/28/09
Read Re: Another AC anomaly?
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
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
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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
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
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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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11/30/09
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mueckenh@rz.fh-augsburg.de
11/30/09
Read Re: Another AC anomaly?
Virgil
11/27/09
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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
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mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
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
Read Re: Another AC anomaly?
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
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/1/09
Read Re: Another AC anomaly?
Dik T. Winter
12/1/09
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mueckenh@rz.fh-augsburg.de
12/1/09
Read Re: Another AC anomaly?
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12/1/09
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Virgil
12/1/09
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mueckenh@rz.fh-augsburg.de
12/1/09
Read Re: Another AC anomaly?
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
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12/2/09
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12/18/09
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12/18/09
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12/19/09
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12/19/09
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12/22/09
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12/22/09
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12/18/09
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12/21/09
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12/21/09
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12/21/09
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12/21/09
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12/22/09
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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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12/16/09
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12/11/09
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12/11/09
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12/11/09
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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
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12/12/09
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12/11/09
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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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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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12/8/09
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12/10/09
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12/3/09
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12/3/09
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12/3/09
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12/7/09
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12/7/09
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12/7/09
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12/7/09
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12/8/09
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12/8/09
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12/8/09
Read Re: Another AC anomaly?
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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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12/10/09
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12/11/09
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12/11/09
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12/15/09
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12/16/09
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12/16/09
Read Re: Another AC anomaly?
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12/16/09
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mueckenh@rz.fh-augsburg.de
12/17/09
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12/17/09
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12/4/09
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mueckenh@rz.fh-augsburg.de
12/4/09
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Virgil
12/4/09
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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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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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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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mueckenh@rz.fh-augsburg.de
12/11/09
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12/11/09
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12/11/09
Read Re: Another AC anomaly?
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12/11/09
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12/11/09
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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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mueckenh@rz.fh-augsburg.de
12/19/09
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12/19/09
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12/19/09
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Virgil
12/19/09
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12/27/09
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12/27/09
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12/27/09
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12/27/09
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mueckenh@rz.fh-augsburg.de
12/13/09
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12/4/09
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12/4/09
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mueckenh@rz.fh-augsburg.de
12/4/09
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12/4/09
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mueckenh@rz.fh-augsburg.de
12/4/09
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12/5/09
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mueckenh@rz.fh-augsburg.de
12/5/09
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12/5/09
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mueckenh@rz.fh-augsburg.de
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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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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