Finite and Infinite OrdinalsDate: 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/ |
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