Does Infinity Exist?

Date: 11/15/2001 at 03:24:49
From: Muhammad Arshad
Subject: Infinity

We talk about infinity all the time, and use it as well. What proof do 
we have that infinity actually exists? Even our universe is said not 
to be infinite, but for convenience we sometimes assume it to be, 
where applicable...

Date: 11/15/2001 at 12:29:22
From: Doctor Peterson
Subject: Re: Infinity

Hi, Muhammad.

What do you mean when you say that infinity exists?

In mathematics, we do not treat "infinity" as a number, though it is 
often used that way informally. Rather, we say that something is 
"infinite," which simply means that it is unbounded or unlimited or 
unending. (That is what the Latin word from which we get the English 
word means.) There is no object called "infinity" that you can point 
out.

What you are really asking is, how do we know that any particular 
entity has no end? The answer depends on the entity you are talking 
about. In mathematics, we deal only with abstract entities like the 
whole numbers or integers, and by definition those are infinite, just 
because there is nothing in our definitions that makes them stop. This 
doesn't mean that all of the infinitely many possible numbers can ever 
be used; on the contrary, it says that no one can ever use up all the 
numbers, even if time does go on forever.

Similarly, in geometry we are dealing with an ideal plane or space 
that has no limits; this has nothing to do with whether actual space 
ends, or curves back on itself, or whatever. For practical purposes 
even in physics, we usually think of space as flat and unbounded; but 
if you are working on a problem in relativity where you go beyond 
small spaces and speeds, you have to start taking curvature and 
possible limits into account. So infinity in physics usually results 
from ignoring parts of reality in order to have a manageable 
abstraction to work with; in math it results from not caring about the 
real world itself in the first place.

So how do we prove that "the infinite" actually exists? It exists _by 
definition_, but only in my mind. "I think it, therefore it exists." 
If you want to know whether anything in the _real world_ is infinite, 
you have to ask a physicist - or, better, a theologian, since I don't 
think science could ever prove that anything is true beyond the bounds 
of what we can observe.

See these Dr. Math FAQs for more on these ideas:

   Large Numbers and Infinity
   http://mathforum.org/dr.math/faq/faq.large.numbers.html   

   Dividing by Zero
   http://mathforum.org/dr.math/faq/faq.divideby0.html   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/