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Finite and Infinite Ordinals


Date: 06/19/98 at 22:49:09
From: D.J.
Subject: Infinite Multiplication

Can you multiply infinities?  I know that you can add infinity to 
infinity to get infinity, but what happens when you multiply infinity 
times infinity?  And could you please tell me some other properties of 
infinity?

Thanks.


Date: 06/22/98 at 13:33:52
From: Doctor Tom
Subject: Re: Infinite Multiplication

Yes, you can multiply infinity by infinity, but you have to be very 
careful.

You can't just slap the number "infinity" into your collection of
integers or real numbers without saying exactly what you mean by
infinity. For purposes of doing mathematical operations, you probably 
want to consider the set of ordinal numbers. You can read about them 
in books on set theory.

The trouble is that there is not just one infinite ordinal - there are 
infinitely many of them. The smallest, usually written as the Greek 
letter "omega," is usually defined to be the set consisting of all the 
finite ordinals:

omega = {0, 1, 2, 3, ...}

The finite ordinals are defined as follows:

0 = {}          - the empty set
1 = {0} = {{}}  - the set containing the empty set
2 = {0, 1}
3 = {0, 1, 2}
4 = {0, 1, 2, 3}
and so on.

There are rules for addition and multiplication of the ordinals, and 
these are described in books on set theory. The one rule that makes 
ordinals more interesting than just the integers is that the union of 
any number of ordinals is an ordinal - that's how you got to omega.

So you can also have:

omega+1 = {0, 1, 2, ...} U {omega}  -- "U" means "union"
omega+2 = omega+1 U {omega+1}
...
omega*2 = omega U {omega, omega+1, omega+2, ...}
...

You can get to omega*omega, and on, and on, and on.

But beware - some common rules that you're used to don't hold:

omega+1 is not equal to 1+omega
omega*2 is not equal to 2*omega

It's a big, complicated subject and there are books written on
it but this should give you some idea of what's involved.

-Doctor Tom,  The Math Forum
 http://mathforum.org/dr.math/   
    
Associated Topics:
High School Sets

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