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Date: Wed, 3 Jul 1996 00:15:34 +0100
From: Nicolas Weibel

Hi Dr. Math!

I'm looking for the demonstration of the Russell Paradox (there
is no ensemble of all ensembles).

Thanks a lot...
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Date: Fri, 19 Jul 1996 16:11:20 -0400 (EDT)

Nicolas,

one of my professors gave an excellent example of the paradox.
Here it is...

Think about a set A, with an interesting definition: A is the
set of all the things that aren't in A , i.e.,

A = {x : x not in A}

Now, consider some element, e.  Is e in A or not in A?

Let's say e is in A.  Then by the definition of A, we must
conclude it is not in A, since A is only made up of elements
that aren't in it. So it can't be in A.  But if e isn't in A,
then by definition, e is in A. So e can't not be in A.

That's the paradox, at least how it was introduced to me.
I know it may be tough to follow this, it is a much easier
thing to explain in person, but I hope this helps.

You can find a slightly different formulation of the paradox,
along with Russell's recollection of how he came up with it at

http://www.cut-the-knot.org/selfreference/russell.shtml

If you have any other questions or want any clarification,
feel free to write again!

-Doctor Erich,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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