Russell's Infinite Set Paradox

Date: 03/25/98 at 18:37:18
From: George McAllister
Subject: Russell's 'Infinite Set' Paradox

Hey Dr. Math,

Can you please explain this paradox to me? All I have read on it 
uses examples like:

Given the set (S) of all sets which do not contain themselves, 
does S contain itself?

The contradiction easily follows: If S does not, then it does, etc. If 
it does, then it does not, etc. My problem is that I cannot imagine a 
set that does not contain itself. If we were to place a set next to 
itself, and compare element by element, then we would see that every 
element in S is in S. I should think that this is simply a definition 
of equality. Maybe it's simply my lack of knowledge. Thanks!

George

Date: 03/27/98 at 13:13:22
From: Doctor Daniel
Subject: Re: Russell's 'Infinite Set' Paradox

Hi there,

You asked about the Russell paradox, which is easily stated as:

     Given S = {X s.t. X is not in X}, is S in S?

If S in S, then, since S in S, it is such an X, and hence S not in S.
If S not in S, then it is not such an X, and hence S in S.

Your question was:

>     My problem is that I cannot imagine a set that does not contain 
>itself. If we were to place a set next to itself, and compare element 
>by element, then we would see that every element in S is in S. I 
>should think that this is simply a definition of equality.

Not quite. We say that two sets, A and B, are equal if all elements of 
A are elements of B and vice versa. But that doesn't mean that A is 
an element of B. 

Maybe an example will make this clear. I apologize for what will sound 
silly, but this is important.

My backpack currently contains my lunchbox, which has an apple and a 
carton of milk in it. My officemate's identical lunchbox contains an 
apple and a carton of milk. Suppose that the apples and milk are 
somehow 
identical.

Then you'll presumably agree that my lunchbox is equal to my 
officemate's lunchbox. After all, their contents are exactly the same.

However, my backpack is NOT equal to my officemate's lunchbox. My 
backpack has a lunchbox in it; my officemate's backpack does not.

For that matter, my lunchbox is not equal to my backpack, for exactly 
the same reason. Also, my lunchbox is not inside my lunchbox. An apple 
and a carton of milk are!  

Here's all of what I just said, in set theory:

     MyPack = {MyLunch}
     MyLunch = {Milk, Apple}
     TomLunch = {Milk, Apple}
     MyLunch = TomLunch
     MyPack not = TomLunch, since MyLunch is in MyPack, but MyLunch
                  is not in TomLunch
     MyPack not = MyLunch, since MyLunch is in MyPack, but MyLunch
                  is not in MyLunch
So:
     Apple is in MyLunch
     Milk is in MyLunch
     MyLunch is not in MyLunch.

I hope that thinking about these sort-of physical examples makes "what 
is a set" more clear. There's no reason why sets can't hold other 
sets, as when my pack holds a lunchbox. The Russell Paradox basically 
comes from the fact that (we know now) sets are not allowed to 
eventually hold themselves. So, for example, my lunchbox is simply not 
capable of holding itself.  

More subtly, I'm not allowed to say something like:

     A = {B}
     B = {A}

Which (again, using the physical analogy) is like putting my backpack 
in my lunchbox, and my lunchbox in my backpack.  

It's good to spend some time thinking about these problems. If nothing 
else, they're fun. You might also check out the Dr.Math archives, and 
maybe visit your library for books about logic.

Have fun!

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