Infinite SetsDate: 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 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/ Date: 09/26/97 at 22:52:41 From: David Gibbs Subject: Re: Infinity Thanks, Dr. Math! We really appreciate it! |
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