Taking the MeasureBy Bobby Neal Winters
In mathematics, simple concepts can be quite subtle. Very early, children take out rulers and begin to measure objects. A foot ruler will suffice to measure a piece of gum; a yardstick is up to the task of measuring one’s forearm; a tape measure will measure a room. None of this is difficult, though it might become tedious if one worries about accuracy, but that is not my concern.
Hey, I am a pure mathematician; what can I say?
Let’s leave the so-called practical concern behind. Consider taking a step from the realm of the concrete to the abstract, from the physical world to the world of pure number.
For example, how would we measure the length of an interval on the number line itself. This isn’t difficult either. If we consider the length of the interval of real numbers [1,3], we can recognize immediately that it is two units long. We figure this out because 3-1=2. It’s simple. Indeed, if we want to make a more general formula, the interval of real numbers [a, b] is b-a units long. Only subtraction is involved.
You man have noticed that I am using the notation that [a,b] is the set of all real numbers between a and b, including a and b themselves. If I write (a,b), this means I am excluding both a and b; (a,b] means I am excluding a; [a, b) means I am excluding b. And all of these intervals are the same length because they differ only by one or two points. Recall that, as far back as the time of the Greeks, Euclid was saying “A point is that which has no part,” so one or two (or three or fifty) points removed from an interval have no effect on the total length. Indeed, one may remove any finite set of points from the interval [1,2] and the result will still be two units long. We aren’t sweating the small stuff.
The sharp reader--and I know you are out there--may have noticed during the course of that last paragraph that I slipped from the concept of the length of an interval to the length of a set. That’s one small step for a man, but one giant leap for Mankind; or Womankind; on Man-unkind if you are e.e. cummings.
In any case, that one small slight of hand opens a whole different kettle of fish. In order for the general reader to appreciate the problem, I will now spend a little while talking about a simple set of numbers that many otherwise gentle people hate with a fury: fractions!
Fractions, I will be talking about them, are better known as rational numbers. As I am no longer talking about this on a street corner or in a bar--pssst,wanna take a look at my square root?-- I’d best call things by their scientific names. Rational numbers are numbers of the form m/n where m and n are both integers but n is not zero. (Remember, dividing by zero makes green hair grow on your palm. Er...never mind.) I mentioned earlier that many otherwise gentle people hate rational numbers: mathematicians love them. Mathematicians are rather like cops, soldiers, morticians, grave robbers, or other people whose jobs force them to see things other people don’t have to, and they have a different standard of nastiness. Compared to some of the critters we’ve seen, rational numbers are as tame as puppies.
One thing that most people don’t know is that rational numbers are literally everywhere on the real number line. Between any two real numbers there is a rational number. This is easy to see; those of you who aren’t up to a dose of algebra can skip the next paragraph.
Take any two real numbers a less than b. We know that b-a is positive. One can choose a positive integer n to be large enough so that n times b-a is greater than one. This means that there is an integer m that lies between n times a and n times be. It follows that m/n is between a and b.
Mathematicians have a name for this property of the rational numbers. We say that the rational numbers are dense in the real numbers. Mathematicians are also frequently referred to as being dense, usually by their spouses, but that is a matter for another day.
The Real Thing
I’ve been referring to the real numbers as well as the rational numbers, and the place that they live together in peace and harmony as the number line. When I was young and full the brashness rightly common to all newly minted PhDs, I understood all this stuff. As I’ve had a been of that brashness scuffed off me, I’ve lost much of that understanding. I try now to approach numbers with a child-like humility. All of this to say that I am now going to explain some things that bright people like yourselves already know.
All rational numbers are real numbers, but not all real numbers are rational: pi is irrational; the square root of two is irrational. Those are two examples, but there are lots of others. Most numbers are irrational. When I say that, I could mean any number of different things. One thing would refer to the concept of number. The set of rational numbers is countably infinite, but, by way of contrast, the set of real numbers is uncountably infinite. I’ve talked a bit about this in other essays. Another thing that I might mean is that the real numbers are infinitely long and the rational numbers have a length of zero.
In case you are surprised that the rational numbers have a length of zero, you are not alone. Indeed, one of the early methods of attempting to measure sets called “content” measured the rational numbers as being infinitely long as well. There were problems with content, however.
Piecing It Together
The problem with content can be easily explained. The content of the rational numbers between zero and one is one; the content of the irrational numbers between zero and one is one. Now, when I take something that is a foot long and stick it together with something else that is a foot long, I don’t expect the result to still be a foot long; I expect it to be two feet long. It is at this point that we can start making excuse for poor little content, but that would take me too far afield. Suffice it to say that content just doesn’t measure up. (If you missed the pun in the last sentence, I urge you to go back and re-read it. I am very proud of it.)
We want the sum of the length of disjoint sets to be equal to the length of all of those sets put together, and we wan that to work even when we join together a countably infinite number of sets. A fellow named Lebesgue figured out how to do this and it works. He cleared up a number of problems that we were having with some very technical mathematics at the same time. We mathematicians love those among us who clear up technical problems. We tend to name things after them and teach courses about those things.
I teach a course about what Lebesgue did. It takes me quite a while to get all of the technical details lined up for the graduate students who take it. I could dump that all on you right now, but you might whimper; my grad students do.
Rather than may you whimper, let me show you something cool that was opened up by our old friend Lebesgue.
Once upon a time, there was an Italian mathematician by the name of Giuseppe Vitali. (Let’s just call him Joe, okay?) Joe created a set that not even Lebesgue could measure. I will show you the set before I let you go. Seriously, if you haven’t noticed, I’ve trapped you in your seat now. You are mine! You are mine I say! Bwahahaha!
Suppose that you could create a set V with the following properties:
- V is contained in the interval [0,1);
- There are sets V1, V2, V3, … each geometrically congruent to V such that
- they are disjoint from each other;
- there union contains [0,1);
- there union is contained in [-1,2).
One might innocently look at those conditions and not think much of it. It all seems so utterly reasonable. Well, that just goes to show what you know.
Suppose we call the union of all of these sets U. As union begins with U this sort of works. On one hand, since U is contained in [-1, 2) and that interval is no longer than three units long, we know that U is no longer than three units long. On the other hand, since U contains [0,1) and that interval has a length of one unit, we know that U is at least one unit long.
So far, so good. It is about to get messy.
Now if things add up the way they are supposed to, the length of U should be the sum of the lengths of V1, V2, V3, … , right? As each of these sets is congruent to V and they are all disjoint, this should be easy. We are just adding the length of V to itself an infinite number of times, right?
Again, the alert reader’s eyebrows may just have arched in a Vulcan-like fashion. This is problem that even those pesky Greeks knew about. If you add up the same number (non-negative) an infinite number of times, the sum is either zero or infinity. It’s zero if the number you are adding together is zero, and it’s infinity if is positive.
So if we there is a set V that has the properties of the of one described, either one is less than zero or three is greater than infinity, or, at least, so it seems.
The faint hearted would stop at this point, but not our Joe.
Joe created V just as we describe above, and he did it in stages.
Joe first broke [0,1) into some disjoint sets. The first of these was the set of all rational numbers in that interval. The rest he obtained by taking copies of the rational by sliding them and rotating them through [0,1).
What’s that? A hand went up in the back. What do I mean by sliding and rotating?
By sliding I mean that you act as if the rationals are marked on a clear ruler the is laying over [0,1) and then sliding that along to cover other numbers. By rotating, I mean that when we push the numbers past 1, act as if the interval in bent in a circle back on itself and those numbers pop up past 0.
“It’s all on a circle, Grasshopper.”
Anyway, once you’ve broken the interval [0,1) into pairwise disjoint sets in that way, you choose one number from each of those sets and form V from those points.
The studious drudges who occupy the ranks of the mathematical world can verify that V satisfies the demands that we’ve put on it.
We are left with a crisis.
Resolving the Crisis
I’d said above that we were left with the choice that either one is less than zero or three is greater than infinity. I left out an alternative. That is that there are just some sets that can’t be measured. The set V is such a set.
In solving problems, mathematicians often find more than they bargained for. This is such a case. Above, quite innocently, I told you to form V by just picking a point from each one of the disjoint sets described. That particular action is now referred to as the Axiom of Choice. It is a method of creating sets that people hadn’t given too much thought to up until that point.
They started thinking about it.
I may as well, but I will be like Scarlett O’Hara and worry about it another day.