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Infinities

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Date: Mon, 21 Nov 94 13:08:00 EST
From: Binford M.S. Gifted Program
Subject: problem

Dear Dr. Math,

Can one infinity be larger than another?

Sincerely,
Gina Miller
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Date: 21 Nov 1994 19:27:52 GMT
From: Ken "Dr." Math
Organization: The Math Doctors
Subject: Re: Gina's problem

Hello there, Gina!

Great question.  Yes, one infinity can be larger than another, a weird
kind of concept.  One standard way of dealing with the different sizes of
infinities is to take one kind of infinity as a "normal" infinity or a
"base" infinity, and then compare its size with other infinities.  What we
do in practice is take the number of counting numbers 1,2,3,4,5,... as our
normal infinity, and we call it "countably infinite."

The way we compare the sizes of two infinite sets is to see whether we can
pair the elements of the two sets up in a one-to-one correspondence.  For
instance, if we match up the even numbers and the odd numbers like this:

2,4,6,8,10,12,14,16,18,...
1,3,5,7, 9,11,13,15,17,...

We can see that there are exactly the same number of even numbers as odd
numbers.  What if we try to match up the even numbers and the counting
numbers?

2,4,6,8,10,12,14,16,18,...
1,2,3,4, 5, 6, 7, 8, 9,...

Again, we can do it, so there are exactly the same number of even numbers
as counting numbers, so we say there are a "countably infinite" number of
even numbers.

Notice that all I've done above is write the even numbers in a list,
making sure that I list them all.  This is the usual way to show that
there are countably many (a synonym for countably infinite) numbers in
some set: try to find a way to list them all.

What about the rational numbers?  Remember that the rational numbers are
numbers that can be written as the quotient of two integers.  I'll write a
list of all the positive rational numbers; can you find the pattern?  If
you're really ambitious, you could try to find a formula for the n'th
number in the sequence!

1/1, 2/1, 1/2, 3/1, 2/2, 1/3, 4/1, 3/2, 2/3, 1/4, 5/1, 4/2, 3/3, 2/4, 1/5, ...

Here's a hint about how to look at the numbers in that sequence: look at
the numerators seperately, then the denominators.  Do you believe that
I'll get all the positive rational numbers this way?  If so, I've just
shown that there are countably many positive rational numbers.  How could
you use a similar sequence to show that there are countably many rational
numbers (both positive and negative)?

It is an interesting fact that you cannot write such a listing of all the
real numbers (which includes both the rationals and the irrationals).  If
you're interested, write back and I'll help you show that.

If that's true, what does that say about how many real numbers there are?
There's a lot of them, more than the number of counting numbers.  So there
are two infinite sets that really do have two different sizes.

I hope this helps you.

Ken "Dr." Math
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Date: Mon, 22 Nov 1993 10:12:04 -0500
From: Phil Spector
Subject: Re: problem (Gina Miller's)

Gina -

Great question!  Infinity is a difficult thing to understand, but if it
weren't there, then there probably wouldn't be many classes left for math
majors to take in grad school.  In any case, depending upon what your
definition of larger is, the answer is yes.  There are a bunch of ways of
looking at this.

If there's one thing that people are sure of, it's that infinity clearly
doesn't always = infinity.  infinity - infinity doesn't equal zero, and
infinity/infinity doesn't equal one.  Rather, they are both undefined,
because it is recognized that one of the infinities could be much larger
than the other.  How?

Well, let's start counting from 1.  You could go on for ever.  In fact, I
think it's safe to assume that since you could count forever, there are an
infinite number of numbers in that set.
In other words, {1, 2, 3, 4, 5, 6....} and so on has an infinite number of
numbers in it.

But then what about {1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5....} and so on?  This
seems to have twice as many members (can you see why?), but it still of
course has an infinite number of numbers in it.

Finally, what if we were to look at every single number you can think of.
That includes all the above numbers, plus a lot more.  And of course there
are an infinite number of numbers you can think of.

Therefore we see that in each case there were an infinite number of
numbers, but in each case there were more numbers than in the previous one.
It seems that indeed, one infinity can be bigger than another.

This is, of course, very silly.  Even though there are twice as many
numbers in the second set as in the first, mathematicians have never gone
off the deep end and said that 2 * infinity = infinity.  Rather, they make
efforts to differentiate between different kinds of infinity.  For example,
they call the first set (where you can count the infinity) a "countable
set" of infinite numbers, while they call the third example (all the
numbers you could ever think of, including all sorts of wacky decimals) an
"uncountable set".

I'm sure some other Dr. Math's will jump in with other ways of looking at
it, but I like the above one, if for no other reason than it's the only one
I could think of.

Hope this helps a little!  Again, great question!  You asked about
something that has been plaguing mathematicians for a long, long time...

Phil, Dr. Math
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Associated Topics:
Elementary Infinity

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