Russell Paradox

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

Hi Dr. Math!

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

Thanks a lot...

Date: Fri, 19 Jul 1996 16:11:20 -0400 (EDT)
Subject: Re: Russell Paradox

Nicolas,

If you're talking about the same Russell paradox that I know, 
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/