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### Infinite Sets

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Date: 09/24/97 at 23:57:54
From: David Gibbs
Subject: Infinity

In my algebra class we have been debating whether the integers or the
whole numbers contain more elements. Some people think the integers
would contain more since there is infinity at both ends. Others
maintain that whole numbers contain just as many elements, since they
are infinite as well, and that it does not matter that they are
infinite only at one end. Can you help us end this argument?

Thanks,

David Gibbs, Math Teacher
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```
Date: 09/25/97 at 06:12:29
From: Doctor Pete
Subject: Re: Infinity

Hi,

As you and your students have seen, our common-sense notion of "size"
does not apply itself so well to things which are infinite.

On the one hand, the whole numbers W = Z+ U {0} = {0,1,2,...} are a
proper subset of the integers, Z = {...,-2,-1,0,1,2,...}, because
every element in W is also in Z, but there exists at least one element
in Z that is not an element in W. So you can say in some sense that
Z is larger. On the other hand, a one-to-one correspondence exists
between elements in W and elements in Z; in particular, the function

F : W --> Z,

{ (x+1)/2,  x odd
F[x] = {    0   ,  x = 0
{  -x/2  ,  x even

maps all elements in W to a unique element in Z, and the inverse
F^(-1)[x] maps all elements in Z to a unique element in W. So in this
sense, W and Z have the same size!

Now, we have seen two very compelling arguments that lead us to an
apparent contradiction: the first tells us that |W| < |Z|, because
there exist elements in Z which are not in W, but the second tells us
|W| = |Z|, because you can pair up every element in W to an element in
Z without any left over in Z. Which is correct?

It is the latter argument which is correct, because the proper
description of the relation between W and Z is that W is a proper
subset of Z, but because both W and Z are infinite sets, this does
not necessarily mean that Z is larger than W.  In fact, one cannot
"compare" sizes of infinite sets in the same intuitive fashion as
finite sets. All you can say is that |W| = |Z| = Aleph-naught, that
is, their cardinalities are equal, and equal to the cardinality of the
natural numbers, which we call Aleph-naught.

A simpler example would be to compare the sets {0,1,2,...} and
{1,2,...}. Clearly the latter is a proper subset of the former, but it
is obvious that you can pair up every element in one set with a unique
element in the other, and have no extra elements left over in either.
Just map 0 to 1, 1 to 2, 2 to 3, etc. Similarly, the even numbers
2Z = {0,2,4,...} is a proper subset of W = {0,1,2,...} but again, a
one-to-one correspondence exists, hence |2Z| = |W|. But the odd
numbers W \ 2Z = {1,3,5,...} also have the same cardinality as
|W|, i.e., |W \ 2Z| = |W|. So quite paradoxically, the even and odd
numbers taken together are the whole numbers, yet taken individually,
each has the "size" of the whole numbers. That is, |W| + |W| = |W|.
But this is no paradox, because |W| = Aleph-naught, which is not a
number but what we call a transfinite cardinal. Aleph-naught is not an
element of the real numbers, is not finite, and as such, it does not
obey the usual field axioms (addition, multiplication, etc.)

So to put everything in a nutshell, to say Z is larger than W is not
quite correct, for the concept of "largeness" in the finite sense does
not apply to these sets. It is mathematically consistent, and
therefore correct, however, to say W and Z are of the same "size,"
though you should say "The cardinalities of W and Z are the same and
equal to the cardinality of the positive integers, Aleph-naught."

-Doctor Pete,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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```
Date: 09/26/97 at 22:52:41
From: David Gibbs
Subject: Re: Infinity

Thanks, Dr. Math!  We really appreciate it!
```
Associated Topics:
College Logic
High School Logic
High School Sets

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