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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


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


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/   
Associated Topics:
College Logic

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