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### Does Infinity Exist?

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Date: 11/15/2001 at 03:24:49
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...
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Date: 11/15/2001 at 12:29:22
From: Doctor Peterson
Subject: Re: Infinity

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/
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Associated Topics:
High School Number Theory

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