What is larger -- the set of all positive even numbers, or the set of all positive integers? We will call the set of all positive even numbers E and the set of all positive integers N.
At first glance, it seems obvious that E is smaller than N, because for E is basically N with half of its terms taken out.  The size of E is the size of N divided by two.
You could see this as, for every item in E, two items in N could be matched (the item x and x-1).  This implies that N is twice as large as E
On second glance though, it seems less obvious.  Each item in N could be mapped with one item in E (the item x*2).
Which is larger, then?  Or are they both equal in size?  Why?
(My background in Set theory is quite extremely scant)

Almost all of this misses the idea that you can't do traditional arithmetic with infinity that way that most of us think of it.  10/2 = 5, five is half of 10, no problem.  If Judy has five apples and Joe has 10 apples, then it is proper to say "Judy has half as many apples as Joe."  But to say "there are half as many even positive integers as there are positive intergers" is to infer something about infinity/2, or half of infinity, which is not entirely kosher.
