(C) 1999 Scott Aaronson

Luminescent green stretches beneath my feet in all directions; above, starless night. I don’t know where I am or how I got here. I scream; no answer. I run; just more luminescent green. At last a minute head appears over the horizon, then shoulders, then a full human figure, hauling some heavy object. The figure draws closer, and I see that it’s a stocky old man with a duffel bag. He’s bald except for tufts of white hair above both ears and his green eyes shine almost as brightly as the surface below. I approach him. "Where am I?" I ask. The old man smiles. "You," he announces in a raspy voice, "are on the back of a giant glowing tortoise." I look at him incredulously. But the old man’s face is sincere, and given the circumstances, I might as well believe him. "Well then," I say, "what’s the glowing tortoise standing on?" "Another tortoise, of course." "And what’s "Another tortoise." "Then I suppose there’s an infinite tower of tortoises, all the way down?" "Right." Pause. "But where "Aha! The "Then what’s the iguana standing on?" "Another iguana." The old man doesn’t need to tell me now that there’s an infinite tower of iguanas all the way down. And the infinite tower of iguanas, as it turns out, is standing on an infinite tower of alligators, and that on an infinite tower of three-headed Tyrannosaurs, and so on, through every kind of reptile, real or fictional, that one might imagine. And how many kinds of reptiles would that be? "Infinitely many." "Then what is this infinite tower of infinite towers of reptiles standing on?" "An elephant." An infinite "And what is this infinite tower of infinite towers of infinite towers of animals standing on?" "A bottle of Tide cleaning detergent." A tower of Tide, resting on a tower of Wisk. And this is but the top of an infinite tower of infinite towers of infinite towers of household cleaning products. "Wait," I say, drawing a deep breath. "We started with this single glowing tortoise. Call that Level Zero. Then we progressed to an infinite tower of tortoises. Call that Level One. Then to a tower of towers of reptiles—Level Two. "A chewed-up piece of gum," replies the old man disinterestedly. I give up. No matter what collection of things I name, the old man will tell me the topmost thing that all of them are standing on. So I change the subject. "What’s in the duffel bag?" "Aha! I’ve just returned from my descent into the transfinite abyss, and I took a photograph of each thing I encountered." The old man pulls a thick photo album out of his duffel bag. "How many photos are in there?" "Aleph-one—uncountably many." Having studied a little set theory, I know what this means. Georg Cantor proved in 1873 that there are different levels of infinity—that, for example, no matter how you pair up the whole numbers with the points on a line, there will always be points left over on the line, even though there are infinitely many of both. In 1891, he generalized this to prove that just as there is no largest number, so too is there no largest infinity—no matter what level of infinity one named, Cantor could name a higher level. A devout Catholic, Cantor believed that this result illustrated the transcendent greatness of God. And though his levels of infinity encountered vehement opposition from such nineteenth-century mathematicians as Leopold Kronecker, today they’re a standard and indispensable tool in math and computer science. The infinity of whole numbers is called Aleph-zero, or ‘countable,’ and is the lowest level of infinity. The next higher level is called Aleph-one, and is ‘uncountable.’ "Of course, you don’t have time to look at uncountably many photos," says the old man, flipping through his album. "So pick one thing—this orangutan, say, or that mouse pad, and only look at the photos of the things above it. Of these there will only be countably many." "But—that’s impossible!" I stammer. I’m thinking, for example, of a line: break off a section of it, and the section still has uncountably many points. "Why so? Look: if I tell you a collection of things underneath us, can you always tell me the bottom-most of those things?" "No—for example, if the collection is the tortoises, then there’s no bottom-most tortoise. They extend infinitely downward." "Correct. But given a collection of things, can you always tell me the I ponder this for a minute, before replying, "Yes. Call the collection C. Then consider all the things that are above everything in C. There must be a top-most thing, call it T, that all of those higher things stand on. Then T is the top-most thing in C." For example, I realize, if C is a collection of reptiles, then there must be a top-most "Now," says the old man, "call the "Well, that there’s a top-most thing in it—we’ve established that that’s true of every collection." "Yes. And what of some thing—a cheese cracker, say—that’s above that top-most thing in the low collection?" Light dawns. "There can only be countably many things above the cheese cracker!" I exclaim. "Because the top-most thing with uncountably many things above it is below the cheese cracker. "Now you see," says the old man. "Any ordered set having the crucial property of the things below us, that for every collection of things there is a single top-most thing, is called a well-ordered set. A mathematician would say that my photo album has order type S-omega—that of the minimum uncountable well-ordered set. And though S-omega is uncountable, for any thing in it there are at most countably many things above it. And that’s why you can choose any photo from my uncountable album and be sure of finding at most countably many photos before it." As I grasp the order governing the transfinite tower of things below me, the luminescent green world seems less strange, less intimidating. But still I yearn to know just "Is there a thing," I ask, "that underlies the whole transfinite tower—upon which stands every collection of things I could possibly name?" The old man falls silent for a minute, before quietly responding, "There is. It’s an immense buttermilk pancake with maple syrup." He anticipates the obvious question. "You’re not allowed to ask what’s beneath the pancake." "Why not? It’s a perfectly valid question." "No it isn’t. The mere fact that you can ask it doesn’t make it a valid question." "Well, why "Because," he intones ominously, "you can never reach the pancake. The only way you’ve found out about the things below you is by specifying collections of things and asking me what they stand on. But whatever collection of things you name, in whatever system of logic, the whole collection will always stand on something well above the pancake. The pancake is the Absolute Infinite—forever inaccessible." The luminescent green world that seemed comprehensible, benign, even delightful only minutes ago suddenly shocks and horrifies me. I want to return home. "How do I get out of here?" "Easily," says the old man. He reaches into his duffel bag, pulls out a handful of dust and throws it into the darkness—and at once the two of us are floating in the center of a sparkling pink dodecahedron. "But this isn’t home!" I cry. "Of course not. The glowing green tortoise was a dream within a dream. I’ve just returned you to the original dream." "Well, get me out of this dream as well." The old man complies, and we find ourselves sailing though a three-dimensional compact manifold embedded in a five-dimensional Euclidean space. I realize that I’m trapped in a dream within a dream within a … all the way down. And if I ask to escape from the infinity of dreams, I’ll only be trapped in another dream—just as an entire infinite tower of tortoises rested on an iguana. You can never escape the transfinite. But then I awake. I gaze at the luminescent green by my bed: 7:26 AM. Hughes Dining is open. Maybe I’ll go get some pancakes. |