On June 4, 1925, the Westphalian Mathematical Society held a conference in honor of Karl Weierstrass. At that conference, David Hilbert gave a famous speech which has since become known as "On the Infinite". In that speech, Hilbert said "...the definitive clarification of the nature of the infinite, instead of pertaining just to the sphere of specialized scientific interests, is needed for the dignity of the human intellect itself. From time immemorial, the infinite has stirred men's emotions more than any other question. Hardly any other idea has stimulated the mind so fruitfully, and yet no other concept needs clarification more than it does." (1)
I could not possibly agree with Dr. Hilbert more. Of all the subjects I teach, none so enraptures my students, and none so infuriates them, as those concerning infinity. Although I have studied the infinite for almost 30 years, I too am endlessly befuddled, inspired, and illuminated by its mysteries. My father first introduced me to the notion of infinity when I was about 8. In the years since, the infinite has been my constant joy and my beloved companion. It is my hope to share some of that with you. I begin where my father did, with the invention of counting.
"Imagine that you are a caveman. Your name, perhaps, is Ugg, and you have seven children, Moot, Shem, Zok, Tov, Boka, Nim, and Tee. You, however, do not know that there are seven, because counting hasn't been invented yet. One day, as you make your way through the wilds, you come upon a tree, heavy with apples. You eat one, and the sweetness explodes in your mouth. You gather some, to take home to your children, but then you pause. As all parents (and all people who have siblings) know, this could be a problem! If there are not enough, the children will fight over them. If there are any extra, the children will fight over them. And yet, how are you to know how many to gather? You sit at the base of the tree and think long and hard, and finally inspiration awakens inside of you. "For Moot" you say, and put an apple in your basket. "For Shem", and you take another apple. "For Zok, for Tov, for Boka and Nim," you recite, putting apples in your basket. "And finally for little Tee!"
You give praise to the Apple Tree, the inspirer of science, and return to your cave, where you produce exactly one apple for each child, much to the wonder of all. In the days that follow, you explain your method to all your people. Days follow days, and the moon grows and shrinks many times. You apply your new method not just to apples, but to days, and to moons. You learn the cycles of nature. You know when the moon will shine and when she will hide her face. You know when the rains will come, when the cold will abate, when the river will flood its banks, and when the apples on your beloved Tree Teacher will ripen. You know how many bison are coming up the river, and how many hunters to take; how long you have to prepare for winter. You become a shaman and a teacher; the number-wisdom has brought previously unfathomable prosperity to your people. The apple tree has taught you many mysteries, but there are ones you cannot fathom. How many numbers are there? Which are there more of, breaths in the sky or drops in a river? Who has more children, the Mother of Grassblades or the Mother of Stones? You are old now. Almost too old to count your years. You sit beneath your beloved teacher one last time, and you begin to number the stars."
This, my father taught me, was the essence of infinity, that it goes on and on forever, unbounded(2) and unbroken. And yet, like Ugg, I wondered if all infinite things were the same size. Which is larger, the set of all counting numbers, or the set of all integers (negative, zero, and positive whole numbers)? Both are infinite, and yet certainly there must be twice as many integers! And so my father (who was a mathematician himself) continued to teach me.
To begin to answer the question, we have to look back at Ugg's insight. The only way we can know for sure that two things are the same size is to set up a one-to-one correspondence between them, like Ugg did between his children's names and the apples. Once we think of it that way, it's quite obvious that the integers (positive counting numbers) and the even numbers are the same size. We match 0:1, 1:2, -1:3, 2:4, -2:5, 3:6, -3:7... Because both the counting numbers and the integers go on forever, we never run out. We have matched every single integer to a counting number, and so just like Ugg knew there were the same amount of apples as children, we can be sure there are just as many counting numbers as there are integers.
What about more complicated kinds of numbers, like fractions? Surely we can't list the fractions like that!
Imagine a grid of fractions, like this:
Not only is every positive fraction on the grid, each is there infinitely many times, since, for example, 1/2 = 2/4 = 3/6 ...
Now, in order to match this grid to the counting numbers, we need to rearrange it into a linear list, which we can do like this (skipping the ones we've already listed):
Here, I proved for you that isn't true; some infinite things are bigger than other infinite things.
I leave you with a short trance contemplation inspired by this lesson:
Take several deep, long, slow breaths, close your eyes, and bring your awareness to the center of your body.
Imagine a string of lights extending from the center of your body up into the heavens. It doesn't matter if you can actually see them or feel them, just imagine that you can. You might want to align the first few with your higher chakras, if you're into that kind of thing. The lights do not end at the top of your head, they continue, up, up, up... past the planets, past the stars, past endless galaxies, to the center of the universe. And yet, even then, they extend, infinite but countable, an endless sequence of lights, twinkling in unison, like the christmas tree of life. :) When you've stabilized this imagining, abide in it a little while, and then return your focus to your center, but allow the lights to remain.
Slowly, one at a time, beginning with the one just above your center, match each light above with a light below, twice the light, and yet no more. The lights go down, down, down, through your feet, through the earth, to the center of the world, and yet they do not end there, they extend, further and further, never ending, never fading, an infinite sequence of lights. When you've stabilized this imagining, abide in it for a little while, and then return your focus to your center, allowing the lights to remain, both above and below.
One at a time, spiraling out from your center beginning with your right hand, parallel to the ground, imagine a vast 2 dimensional grid of lights, parallel to the ground. Match each of these lights with a light on your center-line, infinitely more light, and yet still no more. When you've stabilized this imagining, abide in it a while, and then return your focus to your center, allowing the lights to fade.
Once you're good at this, try filling in the third dimension by raising and lowering your grid so that it intersects every light on the first string.
Eventually, with some practice, you'll get good at that too. Once you can keep the entire three-dimensional world full of lights, add a fourth dimension, perpendicular to the three you already have. The exact nature of this is left as an exercise to the reader. :) (Hint: Start by both spinning each individual light while allowing the entire field to rotate in the opposite direction.)
(1) The talk is somewhat more technical that this quote might make it appear. I'll discuss some of the more complicated aspects in later chapters.
(2) OK, ok, you're right! Something can be unbounded and still finite. Google "elliptical geometry" if you don't believe me.