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### Finite and Infinite Ordinals

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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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