To Infinity and Beyond
Mathematicians sometimes say, “Infinity is not a number.” To the average incurious person, infinity is infinity. It’s simple. There is only one infinity, and it’s very, very big. End of story.
Infinity turns out to be a very tricky, and fascinating, concept. Infinity is anything but simple. Let’s start with the integers. Intuitively, we can see that there seem to be an infinite number of them – that is, the integers just “go on” indefinitely. No matter what integer we think of, we can always add 1, or subtract 1, from it, and get a bigger or smaller integer. Then there are the real numbers. The real numbers include integers, as well as lots and lots of numbers between them. Intuitively, we can see that there are an infinite number of real numbers.
Now since the real numbers include the integers, plus lots of other numbers, it seems that there must be more real numbers than integers. But we just said that there are an infinite number of integers! There seem to be different “amounts” of infinity.
A clever thought experiment, devised by mathematician David Hilbert, also illustrates how tricky the concept of infinity can be. Suppose we have a grand hotel with infinitely many rooms, and every room is occupied. Since the hotel is “full,” it can’t possibly accommodate more guests. Yet if a guest shows up, all we have to do is move the guest from room 1 to room 2, the guest from room 2 to room 3, and so on. Since there are infinitely many rooms, we can continue this indefinitely. Thus the new guest has a room. We could repeat the whole process with each new guest, and accommodate an infinite number of new guests in a hotel that is supposedly full.
In the 19th century, a German mathematician named Georg Cantor became obsessed with the concept of infinity. In the process, he gave us set theory as well as many valuable insights. Cantor realized that 2 sets had to have the same number of objects if there was a one-to-one correspondence between the two. For a finite set, this is pretty straightforward. For example, these 2 sets clearly have a one-to-one correspondence:
But what about sets that aren’t finite? Cantor discovered that even here, we can ask the question, “Do they have one-to-one correspondence?” And we get some surprising answers. For example, let’s take the integers and the rational numbers. A rational number is any number that can be expressed as the ratio of 2 integers. Since any integer can be expressed as the ratio of 2 integers (2 for example is 2/1), it seems that the rational numbers include the integers, plus many more numbers. So it seems that there must be more rational numbers than integers.
Cantor set up a diagram to examine this, with the integers along each side. Each dot on such a diagram represents a pair of integers. Remember that each rational number is a ratio of 2 integers. He then drew arrows, starting at zero, and moving through the diagram. The path of these arrows passes through each pair of integers once and only once. In doing this, we discover something quite remarkable. Each point in the path, corresponding to each pair of integers, can itself be assigned an integer. Each of these integers corresponds to one and only one PAIR of integers. For example, in the chart above, the integer 7 corresponds to the pair (2,1). The integer 8 corresponds to the pair (1,2).
No matter how long we continue this process, we can see that each integer MUST correspond to one and only one PAIR of integers. If we wanted to include the negative integers, we could simply do a second similar chart with them. Again we would find a one-to-one correspondence between the negative integers and the negative rationals. The astounding conclusion is that, contrary to our intuition, there are as many integers as there are rational numbers! Even though both of these sets of numbers are “infinite,” in the sense that they just “keep going on,” mathematicians say that they are COUNTABLY infinite. Cantor used the term transfinite to describe such sets. The number of elements in a set is called the cardinality. Cantor showed us that the cardinality of the integers and that of the rationals is the same. This is represented by the symbol aleph-null.
However, there are many numbers that cannot be expressed as the ratio of 2 integers. The number pi, for example, or the square root of 2. These are called irrational numbers. The rationals plus the irrationals are together considered the set of real numbers. So again, intuitively, there must be more real numbers than rational numbers. Is this correct?
Cantor found an ingenious way to answer this question, again using a chart. Any real number, rational or irrational, can be represented as a decimal, a series of integers. Cantor showed that we don’t even have to look at most of the real numbers to answer the question – only the real numbers that can be represented by sequences of zeroes and ones. Suppose we have series of zeroes and ones like this:
We could go on like this indefinitely, producing strings of zeroes and ones. But notice something. No matter how many of these we generate, we can always generate a series that “flips” the digits on the diagonal (highlighted in red). This new series CANNOT be found anywhere amongst the strings we generate. Each original string MUST have either a one or a zero in a given position. So we can always produce a new string by flipping the ones and zeroes! This simple, incredibly profound proof shows us that the real numbers are UNCOUNTABLE. There are indeed more real numbers than there are rational numbers – rationals are countable, reals are not.
Cantor went further. He showed that the cardinality of the reals (the “number” of reals) has a mathematical relationship to the number of rationals. Remember that the number of rationals is represented by the symbol aleph-null. The cardinality of the reals turns out to simply be 2, raised to the power of aleph-null.
When dealing with these kinds of problems, our intuitions break down. Things that seem contradictory become certainties and vice versa. What is called Russell’s paradox is a good example. In naïve set theory, a set is merely a definable collection of objects. For example, take the set of all sets which do not have themselves as a member. This set seems to contain itself as a member. But by definition, it CAN’T contain itself as a member.
This contradiction comes about because our language is often sloppy. It’s kind of like the statement, “I always lie. I’m lying now.” If I ALWAYS lie, then this statement is a lie. But if the statement is false, then it follows that I’m telling the truth. But this contradicts the statement that I always lie. Ordinary language doesn’t cut it when it comes to rigorous logic. Mathematicians have solved this problem by developing what are called axioms – precise definitions and deductive arguments. In axiomatic set theory, you simply can’t have the kind of set described above. To be contained by one set, a second set must be smaller. Our language and our intuitions can lead us astray. But mathematical rigor still holds – and it leads to some surprising conclusions when it comes to infinity.
We have already seen that there are different “amounts” of infinity – that is to say, some sets are infinite but countable, others are uncountable. What happens if we add infinity to infinity? It turns out, very counterintuitively, that we get the same “amount.” The same is true if we multiply infinity by itself. Clearly, infinity doesn’t behave like a “normal” number.
What is even more counterintuitive is that any interval between 2 real numbers has the same “number” of reals as the ENTIRE set of reals. This is why we call the reals the continuum. For any 2 reals, there are always more reals in between. The “number” of reals between 0 and 0.0000001 is the SAME as the “number” of reals between -1,000,000,000 and +1,000,000,000, which is the same as the total “number” of reals! We are used to thinking of infinity as something big. But infinity also applies to the very small. And this turns out to be quite useful.
Suppose we have a diagonal line. This line has a slope. If we take any 2 points along the line, we can plot their coordinates on a horizontal (x) and vertical (y) axis. If we take the difference between the 2 y-values and divide by the difference between the 2 x-values, we get the slope of the line – how fast y changes as x changes. The slope of a line is constant.
What about the slope of a curve? Well, it turns out that we can determine that too, although it’s a bit trickier. A curve can be thought of as a line that is constantly changing its slope. If we take 2 points on the curve, close together, we can do the same thing we did for the line. This will give us the slope of a line connecting these 2 points. If we take 2 points closer together still, this will give us the slope of a line that approximates the curve more closely at one point. And so on.
Notice that if we take a single point on a line, or a curve, we might say that it doesn’t really have a slope. A slope, by definition, is about a least 2 points separated in space. Right? Well, wrong, sort of, and this is where the concept of the limit comes in. We can understand this intuitively by looking at a circle and a square of the same width. The sides of the square never go inside the circle. They touch the circle at 4 points. Notice that each side of the square touches the circle at only one point. We could rotate the square and all of this would remain true.
Therefore, we can see that, in a sense, each point on the circle DOES correspond to a specific slope. And the same is true of most any curve. At any given point on a curve, we can derive a slope – a function that describes how y is changing as x changes. Welcome to calculus.
Take a line that slopes up to the right. No matter how small a segment of this line we look at, the slope is always the same. We could draw a horizontal line, parallel to the x axis, that represents this – the slope is constant, regardless of the value of x. Even though the line is composed of infinitely many points, we can see that the slope is never zero.
What about the curve below? Its slope is changing. We can select a couple of points on the curve, draw a line through them, and calculate the slope of the line. We can then select another point closer to the first point, draw another line, and calculate the slope. If we repeat this process, we will find that the slope tends to approach a certain value. In other words, the slope of the curve at that point is its value as the 2 points become ARBITARILY CLOSE.
Calculus confronts us with a remarkable paradox – that every point on a curve is associated with a slope, yet by definition, an individual point doesn’t have a slope. The mathematics of calculus are beautiful, elegant, and used every day by scientists and engineers. The concept of the limit, in a way, is like an ingenious sleight of hand – it enables us to “capture” the infinitely small without actually going there.
There is an English word derived from the Latin word numen, but surprisingly few English speakers are familiar with it. That word is numinous. Numinous means profound, awe-inspiring, arousing an intense spiritual feeling. I invite you, dear reader, to explore for yourself the depths of science, philosophy, and mathematics. See if you don’t experience the numinous yourself.