Sizes of Infinities

Date: 01/31/97 at 06:22:45
From: Mark Clackum
Subject: Formula for proving that 1 infinity can be larger than 
another

What is the formula for proving that one infinity can be larger than
another?  The number of fractions between 1 and 1000 should be larger 
than the number of fractions between 1 and 2.

Date: 01/31/97 at 08:52:48
From: Doctor Steven
Subject: Re: Formula for proving that 1 infinity can be larger than 
another

There are lots of ways to prove one infinity is "larger" than another
infinity. For instance, consider the infinite sequence (ak):

    (ak) = 1,2,3,4,5,6,7,8....

Now consider the infinite sequence (bk):

    (bk) = 2,4,6,8,10....

Obviously, (ak) is all natural numbers, and (bk) is all even natural 
numbers.  They both have an infinite number of terms, but (bk) is also
a subset of (ak)!  In fact (bk) looks to have 1/2 the number of terms 
as (ak).  This is called the paradox of infinity and it comes from 
trying to use infinity as a number instead of as an idea.  Comparing 
these sequences you would get 2*(infinity) = infinity.  And in fact if 
you kept going you could eventually prove c*(infinity) = infinity, 
where c is any number. Looking at 2D points compared to 1D points you 
can get (infinity)^2 = infinity.  Keep going in this way and you can 
get (infinity)^n = infinity.

There do exist different types of infinity.  This distinction is sort 
of hard to think about until it's explained.  There are enumerable 
infinities and then there are non-countable infinities.

The sequence of natural numbers is an enumerable infinity; if we sat 
down for an infinite length of time we could count from 1 to infinity.  

The number of points between 0 and 1 on the number line, however, is 
uncountably infinite.  There is no way given an infinite length of 
time we could count the points. Where would we start?  Pick a point 
and I can find another one closer to 0 than the one you picked. We say 
that uncountable infinite items are much larger than countable 
infinite items.  The number of points between 0 and 1 on then number 
line is much greater than the size of the set of the natural numbers.

I hope this helps; infinity is a hard subject to nail down. Here's a 
quote you might enjoy:

"Infinity is like a stuffed walrus I can hold in the palm of my hand.
Don't do anything with infinity you wouldn't do with a stuffed 
walrus."

 -Dr. Fletcher
  Va. Polytechnic Inst. and St. Univ.

(i.e., don't use it as a number)

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