Thoughts on Infinity

Date: 7/19/96 at 3:39:8
From: Anonymous
Subject: Thoughts on Infinity

Hello Docs! I was just reading through your helpful advice when
I came across a question about subdivision of numbers and infinity.
Sydney was commenting on how one infinity could be larger than 
another. I was wondering, since infinity is never-ending (so I think),
wouldn't it be better to say that one infinity can be *denser* 
than another? Maybe I'm wrong, but its an idea. BTW, thanks for
this page; it's really helpful, especially to somebody like me who 
struggles in math.

Michelle
Seattle, WA

Date: 7/23/96 at 14:45:17
From: Doctor Erich
Subject: Re: Thoughts on Infinity

Michelle,

Thinking of one infinity as denser than another infinity is actually 
a great geometric way to visualize the difference between kinds of 
infinities. There are two basic kinds of infinities.  One type is 
called countable, which basically means you can "number" the things 
you are counting.  For example, if you had a bunch of t-shirts and 
wrote the numbers 1, 2, 3, 4, .... on them you would have a countably 
infinite number of t-shirts.  So you can think of the natural 
numbers(numbers like 1,2,3,4,5,...) as countably infinite.  

The other type of infinity is uncountable (mathematicians sure come 
up with creative names...huh?).  Uncountable means there are so many 
you can't "number" them.  This is where infinity starts to get 
strange, but an example of something that is uncountably infinite is 
something like all the real numbers (numbers like 2.34.. and the 
square root of 2).  In fact, there are more decimal numbers between 0 
and 1 than there are natural numbers(1,2,3,4,...)!

Your density idea is a good idea to picture different infinities, 
although my longwinded explanation may have been a lot more than you 
wanted.  Anyway, keep thinking about math...if you want more 
information about infinity or anything else, feel free to write again!  

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