—GEORG CANTOR
Infinity was no longer mystical; it became an ordinary number. It was a specimen impaled on a pin, ready for study, and mathematicians were quick to analyze it. But in the deepest infinity—nestled within the vast continuum of numbers—zero kept appearing. Most appalling of all, infinity itself can be a zero.
Figure 41: Spiraling outward and inward on the plane…
Figure 42:…are mirror images on the sphere.
In the old days, before Riemann saw that the complex plane was really a sphere, functions like 1/x would stump mathematicians. When x goes to zero, 1/x gets bigger and bigger and bigger and finally just blows up and goes off to infinity. Riemann made it perfectly acceptable to go off to infinity; since infinity is just a point on the sphere like any other point, it was no longer something to be feared. In fact, mathematicians started analyzing and classifying the points where a function blows up: singularities.
The curve 1/x has a singularity at the point x = 0—a very simple sort of singularity that mathematicians dubbed a pole. There are other types of singularities as well; for instance, the curve sin(1/x) has an essential singularity at x = 0. Essential singularities are weird beasts; near a singularity of this sort, a curve goes absolutely berserk. It oscillates up and down faster and faster as it approaches the singularity, whipping from positive to negative and back again. In even the tiniest neighborhood around the singularity, the curve takes on almost every conceivable value over and over and over again. Yet as weird as these singularities behave, they were no longer mysterious to mathematicians, who were learning to dissect the infinite.
The master anatomist of the infinite was Georg Cantor. Though he was born in Russia in 1845, Cantor spent most of his life in Germany. And it was in Germany—the land of Gauss and of Riemann—where infinity’s secrets were revealed. Unfortunately, Germany was also the land of Leopold Kronecker, the mathematician who would hound Cantor into a mental institution.
Underneath Cantor’s conflict with Kronecker was a vision of the infinite, a vision that can be described with a simple puzzle. Imagine that there is a large stadium filled with people and you want to know whether there are more seats, more people, or an equal number of both. You could count the number of people and count the number of seats and then compare the two numbers, but that would take a lot of time. There’s a much cleverer way. Just ask everyone to sit down in a seat. If there are empty seats, then there are too few people. If people remain standing, there are too few seats. If every seat is filled and nobody is left standing, then the number of people and seats are equal.
Cantor generalized this trick. He said that two sets of numbers are the same size when one set of numbers can “sit” on top of another set of numbers—one to a customer—with none left over. For instance, consider the set {1, 2, 3}. It is the same size as {2, 4, 6} because we can make a perfect seating pattern where all the numbers are “seated” and all the “seats” are occupied:
1 2 3
| | |
2 4 6
But it’s not the same size as {2, 4, 6, 8}
1 2 3
| | | |
2 4 6 8
because 8 is an empty “seat.”
Things get really interesting when you get to infinite sets. Consider the set of whole numbers: {0, 1, 2, 3, 4, 5,…}. Obviously, this set is equal to itself; we can just have each number “sit” upon itself:
0 1 2 3 4 5…
| | | | | |
0 1 2 3 4 5…
There’s no trick here. Every set is obviously equal to itself. But what happens when we start removing numbers from the set? For instance, what happens when we remove 0? Oddly enough, removing 0 doesn’t change the size of the set at all. By rearranging the seating pattern slightly, we can ensure that everybody has a seat, and all the seats are still taken:
1 2 3 4 5 6…
| | | | | |
0 1 2 3 4 5…
The set is the same size, even though we’ve removed something from it. In fact, we can remove an infinite number of elements from the set of whole numbers—we can delete the odd numbers, for example—and the size of the set remains unchanged. Everybody still has a seat, and every seat is filled:
0 2 4 6 8 10…
| | | | | |
0 1 2 3 4 5…
This is the definition of the infinite: it is something that can stay the same size even when you subtract from it.
The even numbers, the odd numbers, the whole numbers, the integers—all of these sets were the same size, a size that Cantor soon dubbed (aleph nought, named after the first letter of the Hebrew alphabet). Since these numbers are the same size as the counting numbers, any set of size is called countable. (Of course, you couldn’t really count them unless you had an infinite amount of time on your hands.) Even the rational numbers—the set of numbers that can be written as a/b for integers a and b—were countable. By a clever way of assigning rational numbers to their proper seats, Cantor showed that the rationals were an -sized set (see appendix D).
But as Pythagoras knew, the rationals aren’t everything under the sun; both the rationals and irrationals make up the so-called real numbers. Cantor discovered that the set of real numbers is much, much bigger than the rationals. His proof was a very simple one.
Imagine that we’ve already got a perfect seating plan for the real numbers: every real number has a seat, and every seat is filled. That means we can make a list of seats, showing a seat’s number along with the real number that is sitting in it. For instance, our list might look like the following:
The trick came when Cantor created a real number that was not on the list.
Look at the first digit of the first number on the list; in our example, it’s a 3. If our new number were equal to the first number on the list, it would also have a first digit of 3—but we can easily prevent that from happening. Let’s just say that our new number has a first digit of 2. Since the first number on the list starts with a 3 and our new number starts with a 2, we know that the two numbers are different. (This is not strictly true. The number 0.300000…is equal to 0.2999999…, since there are two ways to write many rational numbers. But this is a minor point that is easily overcome. For the sake of clarity, we’ll ignore that exception.)
On to the second number. How can we make sure that our new number is different from the second number on the list? Well, we’ve already determined the first digit in our new number, so we can’t pull exactly the same trick, but we can do something just as good. The second number on the list has an 8 for its second digit. If our new number has a 7 for its second digit, we can ensure that our new number is not the same as the second number on the list; their second digits don’t match, so they aren’t the same thing. We do the same thing on down the list; look at the third digit of the third number and change it, look at the fourth digit on the fourth number and change it, and so on.
Yielding a new number, .27800…, that
is different from the first number (their first digits don’t match),
is different from the second number (their second digits don’t match),
is different from the third number (their third digits don’t match),
is different from the fourth number (their fourth digits don’t match), and so forth.
Going down the diagonal in this way, we create a new number. This process ensures that it’s different from all the other numbers on the list. If it is different from all the numbers on the list, it can’t be on the list—but we already assumed our list contains all real numbers; after all, it was a perfect seating list. This is a contradiction. The perfect seating list cannot exist.
The real numbers are a bigger infinity than the rational numbers. The term for this type of infinity was , the first uncountable infinity. (Technically, the term for the infinity of the real line was C, or the continuum infinity. For years mathematicians struggled to determine whether C was indeed . In 1963 a mathematician, Paul Cohen, proved that this puzzle, the so-called continuum h
ypothesis, was neither provable nor disprovable, thanks to Gödel’s incompleteness theorem. Today most mathematicians accept the continuum hypothesis as true, though some study non-Cantorian transfinite numbers where the continuum hypothesis is taken to be false.) In Cantor’s mind there were an infinite number of infinities—the transfinite numbers—each nested in the other. is smaller than , which is smaller than , which is smaller than , and so forth. At the top of the chain sits the ultimate infinity that engulfs all other infinities: God, the infinity that defies all comprehension.
Unfortunately for Cantor, not everyone had the same vision of God. Leopold Kronecker was an eminent professor at the University of Berlin, and one of Cantor’s teachers. Kronecker believed that God would never allow such ugliness as the irrationals, much less an ever-increasing set of Russian-doll infinities. The integers represented the purity of God, while the irrationals and other bizarre sets of numbers were abominations—figments of the imperfect human mind. Cantor’s transfinite numbers were the worst of the lot.
Disgusted with Cantor, Kronecker launched vitriolic attacks against Cantor’s work and made it extremely difficult for him to publish papers. When Cantor applied for a position at the University of Berlin in 1883, he was rejected; he had to settle for a professorship at the much less prestigious University of Halle instead. Kronecker, who was influential at Berlin, was likely to blame. The same year, he wrote a defense against Kronecker’s attacks. Then, in 1884, the depressed Cantor had his first mental breakdown.
It would be little comfort to Cantor that his work was the foundation of a whole new branch of mathematics: set theory. Using set theory, mathematicians would not only create the numbers we know out of nothing at all, they would create numbers that were previously unheard of—infinite infinities that can be added to, multiplied with, subtracted from, and divided by other infinities, just like ordinary numbers. Cantor opened up a whole new universe of numbers. The German mathematician David Hilbert would say, “No one shall expel us from the paradise which Cantor has created for us.” But it was too late for Cantor. Cantor was in and out of mental institutions for the remainder of his life, and he died in the mental hospital at Halle in 1918.
In the battle between Kronecker and Cantor, Cantor would ultimately prevail. Cantor’s theory would show that Kronecker’s precious integers—and even the rational numbers—were nothing at all. They were an infinite zero.
There are an infinite number of rationals, and between any two numbers you choose, no matter how close together, there are still an infinite number of rationals. They are everywhere. But Cantor’s hierarchy of infinities would tell a different tale: it would show just how little space the rational numbers take up on the number line.
It takes a clever trick to do such an intricate calculation. Irregularly shaped objects can be very difficult to measure. For instance, imagine that you’ve got a stain on your wood floor. How much area does the stain take up? It’s not so obvious. If the stain were shaped like a circle, or like a square or a triangle, it would be easy to figure out; just take a ruler and measure its radius or its height and base. But there’s no formula for figuring out the area of an amoeba-shaped mess. However, there is another way.
Take a rectangular carpet and place it on top of the stain. If the carpet covers the stain entirely, we know that the stain is smaller than the carpet; if the carpet is one square foot, then the stain must take up less than one square foot. If we use smaller carpets, our approximation gets better and better. Perhaps the stain is covered by five carpets of size one-eighth square foot; we would then know that the stain takes up at most five-eighths of a square foot, which is less than our approximation with a one-square-foot carpet. As you make the carpets smaller and smaller, the covering gets better and better, and your total carpet area approaches the true size of the stain; in fact, you can define the size of the stain as the limit as your carpets approach zero size (Figure 43).
Let’s do the same thing with the rational numbers—but this time our carpets are sets of numbers. For instance, the number 2.5 is “covered” by a carpet that includes, for example, all the numbers between 2 and 3—a carpet of size 1. Using this sort of carpet to cover the rational numbers has some very odd consequences, as Cantor soon showed, thanks to his seating chart. That seating chart accounts for all the rational numbers—it assigns each of them a seat—so we can count them off one by one, in order, based on their seat number. Take the first rational number and imagine it on the number line. Let’s cover it with a carpet of size 1. Lots of other numbers are covered by that carpet, but we don’t have to worry about that. So long as the first number is covered, we are happy.
Figure 43: Covering a stain
Now take the second number. Cover it with a carpet of size ½. Take the third number and cover it with a carpet of size ¼, and so forth. Go on and on to infinity; since every rational number is on the seating chart, every rational number will eventually be covered by a carpet. What is the total size of the carpets? It’s our old friend, the Achilles sum. Adding up the size of the carpets, we see 1 + ½ + ¼ + 1/8 +…+ ½n goes to 2 as n goes to infinity. So we can cover the infinite cohorts of rational numbers in the number line with a set of carpets, and the total size of the carpets is 2. This means that the rational numbers take up less than two units of space.
Just as we did with the stain, let’s make the carpet sizes even smaller to get a better approximation of the size of the rationals. Instead of starting with a carpet of size 1, starting with a carpet of size ½ makes the total size of the carpets equal to 1; the rational numbers take up less than one unit of space, in total. If we start off with an initial carpet that has size 1/1000, all the carpets, in total, take up less than 1/500 unit of space; all the rational numbers take up less room than 1/500 unit. If we start with a carpet the size of half an atom, we can cover all the rational numbers on the number line with carpets that, in total, take up less room than an atom. Yet even those tiny carpets, all of which can fit in the span of an atom, cover all of the rational numbers (Figure 44).
We can get smaller and smaller—as small as we want. We can cover the rationals with carpets that, summed together, fit in the size of half an atom—or a neutron—or a quark—or as small as we can possibly imagine.
How big are the rational numbers, then? We defined size as a limit—the sum of the carpets as the individual sizes go to zero. Yet at the same time, we saw that as the carpets get smaller and smaller, the sum of the cover gets tinier and tinier—smaller than an atom or a quark or a millionth-billionth part of a quark—and we can still cover the rationals. What is the limit of something that gets smaller and smaller and smaller without stopping?
Figure 44: Covering the rationals
Zero.
How big are the rational numbers? They take up no space at all. It’s a tough concept to swallow, but it’s true.
Even though there are rational numbers everywhere on the number line, they take up no space at all. If we were to throw a dart at the number line, it would never hit a rational number. Never. And though the rationals are tiny, the irrationals aren’t, since we can’t make a seating chart and cover them one by one; there will always be uncovered irrationals left over. Kronecker hated the irrationals, but they take up all the space in the number line.
The infinity of the rationals is nothing more than a zero.
Chapter 7
Absolute Zeros
[THE PHYSICS OF ZERO]
Sensible mathematics involves neglecting a quantity when it is small—not neglecting it because it is infinitely great and you do not want it!
—P. A. M. DIRAC
It was finally unmistakable: infinity and zero are inseparable and are essential to mathematics. Mathematicians had no choice but to learn to live with them. For physicists, however, zero and infinity seemed utterly irrelevant to the workings of the universe. Adding infinities and dividing by zeros might be a part of mathematics, but it is not the way of nature.
Or so scie
ntists had hoped. As mathematicians were uncovering the connection between zero and infinity, physicists began to encounter zeros in the natural world; zero crossed over from mathematics to physics. In thermodynamics a zero became an uncrossable barrier: the coldest temperature possible. In Einstein’s theory of general relativity, a zero became a black hole, a monstrous star that swallows entire suns. In quantum mechanics, a zero is responsible for a bizarre source of energy—infinite and ubiquitous, present even in the deepest vacuum—and a phantom force exerted by nothing at all.
Zero Heat
When you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science.
—WILLIAM THOMSON, LORD KELVIN
The first inescapable zero in physics comes from a law that had been in use for half a century. This law was discovered in 1787 by Jacques-Alexandre Charles, a French physicist already famous for being the first to fly aboard a hydrogen balloon. Charles isn’t remembered for his aeronautic stunts, but for the law of nature that bears his name.
Charles, like many physicists of his time, was fascinated with the very different properties of gases. Oxygen makes embers burst into flame, while carbon dioxide snuffs them out. Chlorine is green and is deadly; nitrous oxide is colorless and makes people giggle. Yet all these gases have very basic properties in common: heat them up and they expand; cool them down and they contract.
Charles discovered that this behavior is extremely regular and predictable. Take an equal volume of any two different gases and put them in identical balloons. Heat them up by the same amount and they expand by the same amount; cool them down together and they contract in unison. Furthermore, for each degree up or down you go, you gain or lose a certain percentage of the volume. Charles’ law describes the relationship of the volume of a gas to its temperature.
Zero Page 13
