I have always appreciated the power of mathematics but, unfortunately, am not very good at it. Of course, in life all you have to know to succeed is a few more pages in the math book than the next guy.
One thing that impressed me was the notion that there was more than one "infinity". Like most people, I understood infinity to be connected with the number of positive integers. But on various occasions, things seemed not so clear.
One problem is how to determine when two sets of things have the same number of elements (those of you who know math will have to excuse my loose language). The example I use is to think of two sets of things in two separate bags. Suppose I draw one item from one bag and match it to one item from the other bag and I never run out of things in the first bag before I run out of things in the second bag. Well, I think that is a good way to define the number of things in the second bag as no greater than the number of things in the first bag. If, for each thing in the first bag, there is a unique assignment of a thing in the second bag, then I think that is a good way to define the number of things in the two bags as the same.
So, is the number of positive integers less than the number of positive and negative integers? The above rule for determining the answer shows that the number of positive integers is the same as the number of positive and negative integers. All you have to do is start counting the positive and negative integers using only the positive integers. The correspondence goes like this 1 goes with 1, 2 goes with -1, 3 goes with 2, 4 goes with -2, and so on. It is clear that the positive integers will never run out and every positive and negative integer will have a unique positive integer assigned to it. Yes, Virginia, twice the number of positive integers is the same as once the number of positive integers.
It gets worse. Consider the points in the plane characterized by two integers, such as (3,5). Does the number of these points in the infinite plane exceed the number of positive integers? Not if we can count them systematically with the positive integers. And we can. Start at (0,1) on the x axis and travel in a sort of spiral about the origin. First, go to (1,1), then (0,1), then to (-1,1), then to (-1,0), then to (-1,-1), then to (0,-1), then to (1,-1). We have used the positive integers from 1 to 8 to go in a "circle" around the origin. Next move outward to (0,2) and complete the next largest circle. Clearly we will never run out of the positive integers but will end up counting every pair of positive and negative integers.
We would ordinarily think of the number of pairs of positive and negative integers as infinity times infinity. But this exercise shows that our suspected infinity times infinity is just infinity.
So, if we define ordinary infinity as the number of positive integers, is that what we are stuck with? It seems not, because we can find a bag with so many elements that they cannot be counted with the positive integers. The example I am thinking of is the set of infinite decimals lying between 0 an 1 (inclusive).
The classic approach to showing that this set cannot be counted with the positive integers is to first assume that it has been counted and a two column table has been laid out with the positive integers in the first column and the corresponding infinite decimals in the second column. Next, consider the following infinite decimal. Its first digit differs from the first digit of the first infinite decimal. Its second digit differs from the second digit of the second infinite decimal, and so on. Clearly this is a new infinite decimal that nowhere appear in column two of our table. This contradicts the initial assumption that a correspondence between the positive integers and the infinite decimals between 0 and 1 had been achieved.
We ran out of positive integers before we ran out of infinite decimals between 0 and 1.
The number of infinite decimals exceeds the number of positive integers.