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The different sizes of infinity

September 9, 2010

It was a Sunday morning, an appropriate time for the mind to turn to the higher realms, when my friend Bob informed me there were different sizes of infinity. I thought: if this is true — and on subjects like this, Bob isn't someone you're smart to doubt — this is something the world needs to know.

Not everyone will see it that way. Some will say: I can happily go to my grave without ever having achieved a practical grasp of the infinite. Fine, wallow in ignorance. All I know is, when I come to the end of the road and get pitched into the abyss of eternity, I want to have some clue.

Others will say: everybody knows there are different sizes of infinity, everybody being defined as persons who feel they were rooked because they weren't allowed to score above 800 on their math boards. This was basically Bob's take on it. As it happened, the different sizes of infinity weren't his main point — he was merely using the idea as a homely, everyday illustration of a concept that was really complex. When I objected that he'd left me at the starting gate, he cheerfully offered to explain the notion in two minutes or less, then did so in a breezy manner that would probably have had the President's Council of Economic Advisers nodding in satisfaction but as far as I was concerned … well, let's just say my comprehension had conspicuous gaps. I conceded that one could, in fact, have different sizes of infinity. The part I didn't get was how.

It didn't seem right to leave things that way. Geniuses have wrestled with this subject — wrestled with it long ago, in fact; the idea was first propounded by the German mathematician Georg Cantor in 1874. Surely, I thought, it's about time some notion of it filtered down to us. 

Our original gathering having dispersed (we were brunching at my house with friends), I sat down with Bob on the front steps. "Bob," I said, "I hope you don't have anything pressing planned this afternoon. I need to get this figured out."

I did, to my satisfaction anyway. It took 90 minutes. I present the result below. I don't claim it's elegant, but it works for me. Perhaps it'll work for you.  

We start with the basics. The different sizes of infinity we're talking about are those involving numbers. (Note to mathphobes: don't freak. No equations, minimal terminology.)

  1. Take the counting numbers 1, 2, 3 … Everybody understands that there are infinitely many of these.

  2. Now consider the even numbers 2, 4, 6 … Everybody gets that there are infinitely many of these, too.

  3. Some may think that, since the counting numbers are seemingly twice as numerous as the even numbers, the counting numbers are twice as infinite. These people are knuckleheads. You can match up all the counting numbers with all the even numbers, like so: 1 goes with 2, 2 goes with 4, 3 goes with 6, and so on. Every counting number is matched with an even number; every even number is matched to a counting number. Ergo, these infinities are the same size.

  4. You can match up the counting numbers with any set of rational numbers. (Rational numbers can be expressed as the quotient of two integers, provided you don't divide by zero.) For example, consider the set of numbers of the form 1/x — that is, 1/1, 1/2, 1/3 …  These  numbers get infinitely small, and never exceed 1, whereas the counting numbers get infinitely large. Doesn't matter. The two sets can be matched up — these infinities are the same size.

  5. You can match up the counting numbers with many sets of irrational numbers. (An irrational number is any number that, duh, isn't rational. For example, the square root of two, 1.41421356 …, is irrational; the decimal never repeats.) Consider numbers of the form square root of x — that is, square root of 1, square root of 2, square root of 3 … This set can be matched up with the counting numbers. These two infinities are the same size.

  6. You can match up the counting numbers with any other set of numbers that can be generated algebraically — which is to say, through the ordinary operations of mathematics. All these sets are infinities of the same size.

  7. You're thinking: we don't seem to be getting anywhere here. Ah, but we are. Just wait. 

  8. The real numbers, which for our purposes we'll take to mean all numbers, includes some numbers that can't be generated algebraically. These latter numbers are called transcendental numbers. The best known transcendental number is pi, 3.14159265 …  Pi, the ratio of a circle's circumference to its diameter, is a real quantity of obvious usefulness, but it can't be produced algebraically. (There are ways of approximating pi, but that's different.) Since there's no way to systematically generate transcendental numbers, they can't be matched up with the counting numbers and thus can't be counted.

  9. The real numbers, then, include sets of numbers that can be counted and another set of numbers (the transcendentals) that can't. The implication, if you think about it, is that there are more real numbers than counting numbers.  In short, the real numbers are a bigger infinity than the counting numbers. 

  10. This may strike you — it certainly struck me — as cheating. Surely, you think, there's some procedure that would generate all the real numbers and make them countable, from which it would then follow that the counting numbers and the real numbers were the same size infinity.

  11. There isn't. The proof of this was devised by the aforementioned Georg Cantor and makes use of an ingenious diagonal argument. I won't attempt to explain it here, but the gist is that for any possible list of numbers you could devise, there's some additional number that isn't on the list and thus won't be counted. In fact, it can be shown that most real numbers (or better, "most" real numbers, since we're comparing infinite quantities) are transcendental and thus not countable, from which it follows that, as infinities go, the counting numbers don't amount to squat.

  12. Nothing further to add. However, an explanation with only 11 items in it seems profoundly wrong.

That's pretty much it. I acknowledge this proposition takes a while to get your head around, and won't appreciably improve your quality of life. But you can see why I thought you should know.

— Ed Zotti
Photo by Pat O'Neil

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