Infinity in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.
—Georg Cantor, “On Various Standpoints Regarding the Actual Infinite,” 1885
Late one winter afternoon, lanky, gray-haired Jack Bohn lay on the living room couch with his legs propped on two stacked sofa cushions, typing into the worn laptop that rested on his thighs. He was a recently retired mathematics professor, trying to write one more big paper, this one relating to his notion that the natural world is filled with infinities of all sizes. The ultimate goal of his investigations was to reach a conclusion about how the different levels of infinity meshed.
“Alef arthritis,” he threw out to his wife Ulla, busy at her easel across the room, painting one of her glorious landscapes. Jack’s back ached all the time, each day more than before. “Alef arthritis is what ails you?” said Ulla, not overly concerned. “I’ve never heard of it.” She was a graceful woman with a warm, cheerful face.
“Well—I just invented the name. I see alef arthritis as being a stiffness that sets in when matter is cut off from infinity. I have alef arthritis in my back because I’ve lost touch with the transfinite. Stressing about the Planck length.”
“You’re fussing about quantum mechanics again?” said Ulla. She was using her palette knife to craft a spectrum of shades between two blues; a splatter of paint dropped to the floor. They’d learned to live with paint stains on the rug.
“My latest idea is that physical matter is transfinitely divisible,” said Jack. “When my head’s in the right place, I can see it and feel it: levels below levels, down past alef-null, alef-one, alef-two, on and on. But prim, stuffy quantum mechanics is getting in my face, saying that I should bail out at the Planck length scale, which is a piddling ten-to-the-minus-thirty-fifth meters. So lame. So puritanical. What they don’t understand is that the Planck length scale isn’t a wall. It’s a frontier. There’s a whole new subdimensional world below. And it’s intimately connected to the transfinite. That’s what my new paper is about. I’m hoping the physics angle can help solve the Generalized Continuum Problem.”
“That old shoe?” said Ulla with an experienced wife’s friendly mockery.
“The Generalized Continuum Problem is important,” said Jack, beginning to frown. “It’s kind of sad that I’ve worked on it my whole life, and you don’t even know what it is.”
“Explain it to me again, Jack,” said Ulla, sweetening her voice. “Just one more time.”
“You always say that, and then you don’t listen.”
“But I know you love talking about it. And I do like the sound of the math words. They’re so exotic.”
“All right then. Here we go. The different levels of infinity are called alefs, and we number them with subscripts. We start the subscripts with zero, but it sounds cooler to call it null. So the sequence goes alef-null, alef-one, alef-two, alef-three, out through all the alef-k.” As he talked, he gestured in the air.
“My little professor,” said Ulla. She well knew how the alef symbols looked, and she liked their runic shapes. When Jack talked about the alefs, she saw the symbols instead of hearing the words. ℵ0, ℵ1, ℵ2, ℵ3, and ℵk. She also remembered that Jack liked to use his crazy numbers as exponents, like 2ℵ0, 2ℵ1, 2ℵ2, 2ℵ3, and 2ℵk. Whatever that meant.
Just as expected, he continued, “In 1873, Georg Cantor proved that for any k, 2ℵk is larger than ℵk. So 2ℵk might be ℵk+1, or it might be ℵk+2 or something even bigger. Cantor’s guess was that the transfinite numbers are well-behaved, and that 2ℵ0 = ℵ1, 2ℵ1 = ℵ2, and that, in general, 2ℵk = ℵk+1. I myself think Cantor was a shade too cautious. I think 2ℵ0 = ℵ2, 2ℵ1 = ℵ3, and in general, 2ℵk = ℵk+2.”
“And the Generalized Continuum Problem means deciding whose guess is right,” said Ulla, ready to end this discussion.
“Yeah,” said Jack slowly. “Of course both those guesses might be wrong. The general feeling is that the overall pattern ought to be something simple. But proving anything concrete is really hard.”
“I wonder if your back hurts because you won’t stop working on this thing,” said Ulla softly. “You’re retired now, Jack. Why another paper? Look out the window instead. A storm’s coming. Maybe we’ll get some lightning for once. I hope so. I love lightning.”
“I wish I could be more like you, Ulla,” said Jack, setting his laptop on the coffee table and rolling off the couch with an exaggerated grunt of pain. “You’re in touch with the higher infinities without even worrying about proofs. You sculpt smooth shapes from a continuous range of colors. I chop things into symbols and worry about proofs.” He stretched his arms, wincing at the pain in his back. “Dear infinity, please help me.”
The prayer—if prayer it was—echoed in the high-ceilinged room, just now lit by a sudden gleam of sunlight from amid the scudding storm clouds. Jack felt a twitch in his chest. And then he started choking.
He staggered backwards, holding his throat, seeing spots. He bent over and coughed with all his might. Something slid up from his throat. He spit it into his handkerchief. A preternaturally smooth and glassy figure eight. An infinity symbol.
“Are you okay?” asked Ulla, laying a hand on Jack’s shoulder.
“Look,” he whispered, not trusting his voice.
“Ick,” said Ulla, stepping back.
“It’s not gross,” said Jack, gaining confidence. He began polishing the loop with his hankie. “It’s like a crystal or a jewel.”
“You coughed up a tumor? How horrible!”
“Listen to me, Ulla. This is a miracle. I asked infinity for help and infinity came here.” He laid the amulet down on the coffee table; it made a reassuringly crisp click.
Brow furrowed, Ulla leaned closer, studying the crystalline lemniscate, its interior filled with reflections and bright caustic curves.
“I feel dizzy,” she said. “Like I’m leaning off a windy cliff.”
“I think there’s power in this thing,” said Jack.
“What if it’s some kind of bait?” said Ulla. “To draw us into a trap.”
“Wow, it just poked out a little stub,” said Jack obliviously. “A square plug! I bet I can jack it to my computer.”
Ulla wasn’t liking any of this. “Isn’t a computer the opposite of infinity?”
“I’ll let infinity show my computer where it’s at.”
Jack plugged the infinity symbol into his laptop and—the screen went into an endlessly regressing crash sequence of smaller and smaller windows, each one visible for half as long as the one before. Upon completing the series, the system gave a triumphant beep. The screen glowed white and displayed lines of black text.
CPU: Absolutely continuous matter.
Memory: Alef-null bytes activated.
Runspeed: Alef-null cycles per second.
“Score!” exulted Jack. “Can you believe this is happening, Ulla? I’ve thought about this for years. I know just what to do. I’ll—I’ll use my laptop as a Turing Evaluator. That way I can automatically generate my next paper, ‘Physical Applications of Transfinite Set Theory.’ I won’t have to write it at all.”
“And then we can finally take our vacation in the South Pacific,” said Ulla. “We’ll go diving. I’ll make paintings of the corals and the fish.”
“Yeah, baby. And I’ll have fun reading my new results! Here’s the way I’ll do it. I’ve got my other papers in files on my laptop, see. So I can use a simple little program to search through all the possible Turing machine text-generators to find one that generates files identical to my previous twenty-six papers—and then generates a brand-new twenty-seventh paper entitled ‘Physical Applications of Transfinite Set Theory’!”
“You’re doing monkeys on typewriters? That takes forever.”
“Not forever. Less than alef-null steps, if there’s a suitable program to be found.” Jack’s fingers danced across the keyboard. “Like I said, I’ve thought about this before.” As if energized by the presence of the infinity amulet, Jack was working very fast. “All set. Here we go.”
The search was successful. In less than a second, a new file had been saved on Jack’s hard drive. Not bothering to read it yet, Jack sent the paper to his printer in the next room. The machine hummed, pulsing out the pages.
But meanwhile the infinity symbol in the side of the laptop had grown dim. Reflections of the computer plug were filling the crystal’s interior with an ugly grid of orderly reflections.
“The amulet’s not happy,” said Ulla. “Unplug it.”
Jack pulled the crystal from the side of his machine. The shiny loop brightened. Writhing slowly in his hand, the lazy-eight smoothed away its plug, unknotted itself and became a zero.
Floating into the air, the circle grew to the size of a companionway door. And now a pair of figures stepped through, in the midst of a discussion. The heralds from the higher world resembled—
“A pencil stub and a toad?” exclaimed Ulla. “Are we going crazy?”
The pencil stub had white-gloved hands, legs with backwards knees, and eyes like a pair of glasses animated with black dots on white disks. He strutted across the floor, his point alertly aimed at the humans, his pupils tracking their every move.
The toad was taller; he walked on two legs and wore a baggy gray business suit. His bare chest-skin was pearly green with irregular spots of yellow. The slumped lump of his head sported eye bumps and a wide, downturned mouth.
“Hello Jack,” rasped the toad. “We were talking in the Szkocka cafe when we heard your call—and my overexcitable friend here tossed down an infinity-link. He has this crazy idea for getting you two to help with this problem we’ve been debating. The Generalized Continuum Problem.”
“You’re mathematicians?” exclaimed Jack happily. “The Generalized Continuum Problem?”
“Where did they come from?” demanded Ulla, walking around to peer at the back of the hoop.
“Alefville,” said the pencil stub in a clear tenor. “We’re transfinite beings; we call ourselves aktuals. And my full name is—” An intense, skritchy sound filled the room. It was like hearing someone handwrite an endless Library of Babel in a fraction of a second.
Ulla nodded her head appreciatively and fastened on a shard of the sound storm. “You said Stanley?”
“That’ll do,” said the bird-legged pencil stub. “And call my toad friend ‘Anton.’ For antagonistic.”
“Stanley takes everything so personally,” said the toad, spreading the fingers of his webbed hands. “When I tell him he’s a self-deluding dreamer, he doesn’t appreciate that I’m trying to help. As for his plans for you, I’m not really sure that—”
“Oh shut up,” interrupted Stanley. “I’m offering them a free trip to Alefville.”
“We would grow?” said Ulla uneasily. “I don’t want to burst our house.”
“It’s more that you’ll be changing your focus of attention, ” said Stanley, narrowing the ovals of his cartoony eyes. “Basically, you’re already in Alefville. Infinity is everywhere. This portal is just a visualization tool.” He nudged the glowing ring with the sharpened tip of his nose. The ring rotated to a horizontal position and sank down to shin level, bobbing like a hula hoop.
“We’ll hold hands and hop through all together,” said Anton. “And, Stanley, I’m playing red again. I don’t believe you actually have a winning strategy. You’ve fooled yourself again.”
“I’ll keep on beating you forever,” said the cocky pencil stub. “Thanks to my absolute vision of the true class of all sets.”
“Absolute self-delusion,” croaked Anton, blinking his big golden eyes. “There is no great almighty One. Only the pullulating congeries of axiom models.”
“Wait!” said Ulla, looking suspicious. “You’re not taking us to some giant math seminar are you? My idea of hell, for sure.”
“You won’t be gone long,” said Stanley, not quite answering her question. He turned his pointed nose, gazing out their living room window. “You’ll be home for tea when the rain starts.”
In the next room, the printer had stopped. It gave Jack a good feeling, knowing that his new paper was done. Even if—worst case—he never came back at all, his masterwork was finished. “Let’s go for it,” he urged Ulla. “This might be just as interesting as diving in the tropics.”
So the four of them held hands in a circle and hopped through the hoop—willowy Ulla, pencil-stub Stanley, graying Jack, and Anton the toad.
They found themselves high in the air, falling like a star of skydivers. Far below them, an irregularly shaped coastal city sprawled across verdant hills and fields. The pinkish city’s shape seemed vaguely familiar to Jack. Inland, shockingly vast plains were broken by mountains and still more mountains, the slopes and prairies spotted with smudges of towns, the distant peaks piled up to meet dark, lowering clouds. Out to sea, a sun danced above the endless waves, a strange sun like the mouth of a twitching tube. Rivers meandered from the mountains through the rosy city, forking and rebranching beyond all measure. Uncountable numbers of islands crowded the shore.
“I want us to land on the green,” said Stanley, angling his faceted body so that the four moved a bit to the left.
Anton waved a finned foot, sending them a few inches the other way. “Sorry, Stanley, this time we’re landing on red. We’ve got alef-null turns to go, so make it snappy.”
As they dropped downwards, ever more detail hove into view. What looked like a solid tongue of reddish buildings turned out to have a green park within it, but then the park developed a small block of houses that expanded into a whole new neighborhood spotted with still smaller parks—and this kind of transformation happened over and over again.
The rivals alternated moves at an ever-doubling rate, dithering between greensward and pavement. They were accomplishing an infinite task by splitting a one-minute interval into alef-null smaller and smaller parts.
“A Zeno speed-up,” murmured Jack, who’d often pondered the ancient philosopher’s paradoxical observation that any unit is an endless sum of the form 1/2 + 1/4 + 1/8 + 1/16 + etcetera. Every stretch of time held an actual infinity of intervals although, yes, most of these intervals were below the Planck scale. But the Planck hobgoblin seemed to have little force in Alefville.
The sequence of moves converged upon the four companions landing on a dark pink sidewalk between a bushy green-leafed tree and a multistory apartment building. The ornately decorated building bore a chiseled stone title: Graf Georg Arms.
Pedestrians of all shapes were ambling by; cars crept down the street towards the distant sea. Rather than having wheels, the cars were like millipedes, each with alef-null legs.
“I win,” gronked Anton. “We landed on red. So much for your so-called absolute vision, Stanley. I’d say Alefville’s shape is so kinky that there is no unbeatable strategy for our little steering game.”
“No strategy at all?” said the pencil man crisply. “An absolute truth. Interesting assertion, coming from you.”
“Well, I suppose there might be a strategy lurking somewhere far away,” amended the toad man. “Maybe up in the hill cities—who knows.”
“So the higher levels of infinity can affect the low-level sets?” said Stanley, intense and on the attack.
“Look at this tree, Jack,” interrupted Ulla. “The branches are majorly twisty. And the leaves—there’s so many of them that the canopy is smooth.”
Indeed. The tree’s foliage resembled a car’s glossy green fender. Peering under the leaves, Jack observed that each branch had an endless number of jiggles—as many forks as the natural numbers. Every possible path through the twiggy maze ended in a leaf. Incredibly, Jack could distinguish each one of them.
“There’s two to the alef-null leaves,” he murmured. “The cardinality of the continuum. The size of the real number line. The—”
“Teach us, prof!” said Anton.
Stanley sketched the mathematical symbol for the number on the sidewalk: 2ℵ0.
“We don’t need math words and symbols,” said Ulla, running her hand across the leaves. “Not anymore. It’s so nice to see what you’ve been talking about all these years, Jack.” Gently she parted the foliage, savoring the rich textures. “There’s one special leaf on the edge of this bunch,” she observed, waggling a tuft of green.
“You can think of that edge leaf as an irrational number,” said Jack. “Like the square root of two.”
“This is a branch,” said Ulla, teasing him a little. “Not a root.” She let the leaves snap back into a smooth, bulging surface. An endless swarm of gnats floated out of the foliage, twisting in an unsteady column.
“Let’s go to the Szkocka cafe,” said Stanley, gazing up at them from knee level. “You’re going to help us with the Generalized Continuum Problem, remember?”
“How are we supposed to help you?” said Ulla, nudging the oversize pencil stub with her toe. “You’re the one who’s supposed to be so smart.” She was still looking around, taking in their surroundings. “Check out this apartment building next to us, Jack. It has infinitely many stories. And then more stories after that.” She staggered back a little, craning her head, nearly bumping into a passing dog with an endless number of teeth.
Jack stared upwards at the Graf Georg Arms, his mind boggling. The upper floors were exceedingly low-ceilinged, so the Graf Georg wasn’t unreasonably tall. The building started out like a Zeno speed-up—and higher up it got even weirder than that.
As Ulla had said, the stories didn’t stop after a single run of alef-null levels. The count started up again after the first alef-null, and then again, over and over, infinities mounting beyond infinities, all of them fitting into the building’s finite height.
After a minute’s study, Jack realized that there wasn’t any single master Zeno speed-up of eye-twitches that could sweep his view all the way to the top. The building’s floor-count was a higher level of infinity, beyond the reach of any alef-null-long sequence.
As if attracted by Jack’s mental efforts, the sun shifted its position in the sky, aligning itself behind the summit. The light flooded Jack’s eyes, but he felt no discomfort. Everything seemed to turn white; his ears filled with a mighty roar.
And then he was standing atop the Graf Georg Arms, clutching an antenna mast, peering down towards three antlike figures on the sidewalk: the talking pencil, the toad man, and Ulla, who was standing with her back turned, gesturing at something. Tiny Anton fixed Jack with his golden eyes—and Jack was back on the sidewalk as before.
“Yes,” said Anton, as if Jack had spoken a question. “The Graf Georg Arms has alef-one stories. Most of the buildings in Alefville are that size. You didn’t notice that on our way down because the buildings’ upper floors are compressed into the infinitesimal subdimensions. The town itself has alef-two streets, by the way. Otherwise we’d have a terrible traffic problem.”
“I hopped up to alef-one!” gloated Jack, still elated from his quick round trip. “The sun helped me. Did you see, Ulla?”
“Huh?” She was busy playing with the swarm of gnats from the tree. Moving in tune with the motions of her hands, the dots were forming themselves into deliciously curved bronze sculptures, then deliquescing back into disconnected point sets.
“Never mind.” Jack returned his attention to his two guides. He was eager to talk math. “For you aktuals, the bottom-level Continuum Problem is as real as sorting mail! Never mind about the Generalized Continuum Problem for now—just tell me this: Is it possible to take all the leaves off this tree and give each leaf its own room in this apartment building? Is two-to-the-alef-null the same size as alef-one?”
Anton twitched his wide mouth and glanced down at Stanley. “Should we tell him?”
“Of course,” said Stanley, who’d been absentmindedly scratching some private calculations on the sidewalk with the point of his nose. “Give Jack a treat before we put him and Ulla to work.” He turned to scold a snowman-like passerby made of a stack of alef-null spheres. “Watch where you’re bouncing that big bottom of yours!”
“So okay, Jack,” continued Anton. “It turns out that, no, you can’t give each leaf its own room in the alef-one-sized building. Here in Alefville, the size of the continuum is alef-two. You could in fact put each leaf on its own street-corner—remember that we have alef-two streets. And, assuming that you impose some reasonable zoning restrictions, the same thing’s true in every possible version of Alefville.”
“It’d be simpler if Anton would just admit that we’re in the one true Alefville,” said Stanley. “But never mind. In any case, Anton and I agree that the basic Continuum Problem is solved by a reasonable new set theory axiom—whose details I won’t bore you with.” He was still writing on the sidewalk. As always when mathematicians got going, images of the succinct symbols began replacing the sounds of the words. “In any case, we’ve established that 2ℵ0 = ℵ2. And, as it happens, we also proved that 2ℵ1 = ℵ2 as well. And for the next three levels, things flatten out. That is, we proved that 2ℵ2 = ℵ3, 2ℵ3 = ℵ4, and 2ℵ4 = ℵ5.”
“What does ‘two-to-the-alef-four equals alef-five’ even mean?” put in Ulla. “You guys are as bad as my husband.”
“It means that if you had a tree with branches alef-four forks long, you’d get a canopy of alef-five leaves,” said Stanley primly.
“So—you’re close to proving an answer to the entire Generalized Continuum Problem?” said Jack, growing excited.
“Not at all,” said the pencil stub, still writing on the sidewalk. “We’re stuck. We can’t prove anything about 2ℵ5. If I had to bet, I’d take a wild guess that for any infinite cardinal ℵk after ℵ1, we have 2ℵk = ℵk+1, more or less like Georg Cantor expected. There’s only that one anomalous double step at the start, where 2ℵ0 shoots up to ℵ2.”
“I like it,” said Jack. “But—”
“But we can’t think of any good axioms for proving our solution,” continued Stanley. “And we don’t have any good intuitions about it either. That’s why Anton and I were attracted to your notion that there might be a connection between the higher infinities and the physical levels of subdimensional infinitesimals. There could be a sense in which—”
“This is so dull,” said Ulla. She was staring up towards the mountains beyond town. “And it looks like we’re in for a storm.”
“The lightning’s coming to get you, Ulla,” said Anton, as if making a mean joke. “Let’s head for the Szkocka cafe.”
“You can’t scare me with lightning,” said Ulla. “Is the cafe far?”
“The Szkocka is down by the ocean bluffs,” said Stanley. “Alef-two blocks away.”
“How can we walk alef-two blocks?” exclaimed Jack. “I had to merge into the sun just to reach the top of the Graf Georg’s alef-one stories. Let’s find some shelter around here before the storm begins.”
“Has to be the Szkocka,” said Anton as the first rumble of thunder came booming down. “X marks the spot,” he added, giving Stanley a mysterious wink.
“We don’t have to walk,” said the lively pencil stub. “To get there, we meditate upon Absolute Infinity—and realize that we’ve fallen short.”
“That’s his way of saying we merge into the sun,” said Anton with a sardonic twitch of his mouth. The toad man pointed his skinny arm towards the distant, shining sea. “Thataway!” The loose-hanging fabric of his shiny gray suit fluttered in the rising wind.
Although the thunderclouds had darkened the sky behind them, the sun was beaming across the water, beckoning them. Holding hands with his wife, Jack focused on making the alef-two-sized jump. But something was holding them back. Ulla.
“Stare into the sun,” he urged.
“And go blind? No thanks.”
“The light’s gentle,” said Jack. “It fills you up.” The thunder pealed again, an unearthly, drawn-out sound with a chatter of alef-one echoes at the end.
“When do we go home?” said Ulla, sounding less confident than before. “Stanley said we’d be home before the rain.”
“Just a little more exploring, Ulla,” implored Jack. “We may never visit this world again.”
The light of the hollow-looking sun flowed into them like a long drink of milk. Everything grew white. Anton jostled Jack, bringing him back. They were on a seaside bluff beside a stodgy plaster building with towers set into its corners. Lamps glowed in the windows. Violin music, conversation and laughter drifted out—along with the smells of coffee, beer and fried food.
Out to sea, the waves’ crests were glassy green in the setting sun, the scattered islands were rimmed in gold. But the sky directly overhead was a mass of dark curds. The storm clouds had followed the four companions.
With abrupt violence, the rain began. Alef-null, alef-one, alef-two droplets spattered up from the pavement, writhing in fantastic patterns of fog and spray. Jack and Ulla squeezed under the cafe building’s eaves, still holding hands. And now Jack noticed something very disturbing. Someone’s pencil point had scrawled a large X upon the sidewalk precisely where Ulla stood.
Before he could say anything, the lightning struck. A blaze of light, a tingle in his hand, a hideous crash—and Ulla was gone. Not quite deafened by the thunder’s blast, Jack heard Stanley let out an involuntary cackle. He seized the little creature, digging his fingers into the yielding, polyhedral surfaces.
“Don’t hurt me!” shrilled the pencil stub. “She’s on her way to Absolute Infinity! Look up there in the peaks!”
A ragged rent had opened amid the clouds. Jack saw the lightning picking its way up the misty, mounded range, strike after strike. He could see an image of Ulla at each blast’s core, as if she were an ascending, nimbus-wreathed saint. Perhaps the transport was ecstatic for her—but perhaps not. He squeezed the pencil even harder, wanting to snap the sneaky aktual in half.
“The lightning embodies an extreme large cardinal mapping,” jabbered Stanley. “An iterated embedding that’s carrying your wife towards the top. If you just set me down, we’ll help you go after her, I swear.”
Jack glared at the pencil, then let him drop. “I’m ready.”
“Uh, first let’s step inside the cafe and dry off,” said Anton. “There’s no great rush. We’ll be sending you a different way, Jack. Testing your theory! As above, so below. If you shrink far enough into the physical absolute continuum, you should be able to flip viewpoints and meet Ulla. You and your wife are such a closely matched pair that I’m sure you’ll find her—no matter what.”
The rain’s intensity had redoubled some alef-one times in a row by now. Mountains, sun and sea were lost in the howling gale. The hazy sheets of water were manifesting bizarre forms—bricks, bottles, wheels, chains, and literal cats and dogs. It almost seemed as if the cafe building might melt away.
“Come on, now,” urged Anton. Jack had no choice but to follow the two aktuals inside.
The Szkocka held a lovely great fire in a hearth. The burning sticks had alef-null branchings, but the subtler flames had alef-one forks. Looking closer, Jack saw how wood and flame merged into alef-two infinitesimal eddies of smoke.
“I want to find Ulla,” he repeated.
“Relax,” said Anton, making an intricate gesture towards a distant waiter. “I’m ordering us something. Now to find seats.”
A cluster of alef-null easy chairs sat jigsawed around the fireplace, fitting in via odd warps in perspective. But all were occupied, and by a single rude guest, a cuttlefish who was letting his tentacles loll onto each and every one of the chairs as he scribbled inkily in an alef-one-paged notebook.
“Just move each of your limbs three chairs closer to the fire,” Stanley told the cuttlefish.
“No,” snapped the selfish cephalopod. “Too much work.”
“Never mind,” said Anton. “There’s more room over there.”
Pushing past the cuttlefish, they entered a bulging alcove of space which held an unexpected trove of alef-three tables, with alef-one chairs per table. The further chairs were occupied by shifting gauzy beings, group minds like insect swarms, each swarm an alef-two-sized set of dust motes, the particles congealing into bodies by turns fractally rough or immaculately smooth. Jack and his two guides found three chairs near the door.
A waiter resembling a mushroom arrived with a shot of oily spirits for Stanley, a plump grub worm for Anton, and a miniature cup of coffee for Jack. Anton nailed the grub with his tongue, and sat back smiling. Smooth pressure waves of sound were filling the Szkocka’s transfinite yet cozy space: the sociable weave of words, the endless clatter of plates, the sweet notes of strolling violins.
“Ulla would have liked it here,” said Jack, dropping a sugar cube into his coffee. “Poor Ulla. Let’s go for her soon.”
“Slug down your coffee and we’re on our way,” said Stanley, dipping his sharp nose into his murky glass. His dark pupils were like pinpoints in the centers of his flat, white eyes.
With his first swallow of the nasty brew, Jack knew his coffee was drugged. But the knowing came too late. He slumped back into his chair, overcome by helpless lassitude. The last thing he noticed was Anton picking him up and throwing him across his shoulders.
When Jack came to, he and the two aktuals were in a small boat, out past the islands. The bow seemed to be riding unnaturally high. Although the storm had blown away, the sky was dark. The ocean waters glowed, as with phosphorescence. Sitting up, Jack realized that his ankles were bound together by a cord.
“Just in time,” said Stanley. “We’re about ready to send you in search of your wife! Good luck with that. All we want from you is that you keep an eye on the Generalized Continuum values along the way. That’s the point of this exercise, okay? Ready, Anton?”
Anton was straining over something in the stern. “Can’t—lift—weight,” he grunted.
Weight? Sure enough, the crazy aktuals had tethered Jack’s ankles to a massive block of some preternaturally dense substance, a truncated square-based pyramid whose weight was pushing the rear gunwales down to the level of the lambent waves.
“Wait, wait!” shouted Jack. “How can throwing me in the ocean take me up into the mountains?”
“Don’t you understand your own theories?” said Stanley. “The subdimensions are dual to the transfinite. Two ways of looking at the same thing. Once you’re deep enough, just shift your point of view, and you’ll be up on the peaks with Ulla.”
“Here we go,” said Anton, levering the weight upward with an oar.
“Hop lively, Jack!” yelled Stanley. “And hold your breath!”
Instants later, Jack had been yanked beneath the luminous sea. Glancing up at the ocean’s receding surface—like a wrinkled mirror when seen from below—Jack realized that, yes, this sea’s surface was literally the Planck length scale frontier that shrouds the infinitesimal subdimensions. Everything above was dual to what lay below.
Relentlessly the weight dragged Jack into the abyss. He was in the subdimensional zone for true, passing the reciprocals of alef-null, alef-one, and more. And, by thinking in terms of mathematical duality, he could see the small as the large. He drifted past a mauve sea-fan that branched alef-four times, with alef-five polyps waving from the fan’s fringed rim. 2ℵ4 = ℵ5, just as Stanley had claimed.
Deeper and deeper Jack sank, falling past fantastic architectures of undersea cliffs. Whales beat their way past, singing alef-seven-toned songs; sea-monsters gestured with alef-eight arms. Whenever he glimpsed a branching structure, Jack checked the numbers, filing away data about the Generalized Continuum Problem. Contrary to expectations, 2ℵ10 was a bulky ℵ13, although 2ℵ17 was a svelte ℵ18.
So far Jack was having no trouble holding his breath—but he had a sense that he hadn’t progressed nearly far enough to have any hope of matching Ulla’s progress. He almost wished Anton had tied an even heavier weight to him. As things stood, this quest was up to him.
Looking into his own mind, Jack found an inner sun, the very core of his sense of self. He merged into it and shrank to radically smaller levels of the infininitesimal—leaving the pyramidal weight behind. He drifted like an animalcule in the all-pervading deep-sea light. Smaller, deeper, and—aha. A school of subdimensional paramecia were flowing towards a vent. He followed along, wriggling into glittery fissures. He could sense that Ulla was nearby.
Jack consciously flipped from one mindset to the other, turning his surroundings into an icy gray puddle at the lip of a glacier’s crevasse. Gathering his wits, he molded his body into its customary shape, taking a deep draught of the tenuous alpine air. An articulated drone filled his ears. Overhead arched the vault of the sky, an unblemished cobalt dome, curiously low.
Jack picked his way off the glacier and up a scree of stones with inconceivably many facets. At the peak stood a marble palace, looking out upon subordinate ranges of peaks in every direction. Was this the top? Absolute Infinity?
The droning chant was coming from within the temple. Peering in through the portico, Jack saw a host of singers, praising a figure on a high altar: a large eye resting upon a golden platter. Lo and behold, one of the choristers was Ulla, her skin a bit darkened from the lightning bolts, but otherwise in fine form.
“Jack! You found me. I hate this place. That big bossy eye up there, his name is Jayvee. He thinks he’s so great. Get us out of here. I can’t figure out how to get down.” Her gentle voice was breathy in the thin air.
Jayvee twitched as Jack ushered Ulla out. The big eye didn’t want to lose his new recruit. His golden platter rose into the air, and he followed the couple onto the temple’s porch. The eye extruded a snaky arm bearing a flaming sword. Jayvee was preparing to smite them.
“Dear infinity, please help me,” said Jack once again. The sword swung, Jack and Ulla ducked, and a faint pop sounded from high above.
In an instant, the blue dome became crazed all over with cracks—and fell apart, revealing a much higher range of mountains, stretching up towards a mighty sun. The bullying Jayvee quailed and took refuge in his little temple.
“We are not going any higher!” exclaimed Ulla, guessing Jack’s thoughts.
Jack and Ulla went tobogganing down the slick curves of the glacier’s crevasse—and as they neared the base, they flipped to the dual view of things, becoming subdimensional plankton in the glowing sea’s benthic abyss. From here they drifted effortlessly upwards.
As they rose, Jack watched the branching marine forms even more closely than before—and reached his own conclusion about the Generalized Continuum Problem: the powers of the successive alefs obeyed no uniform pattern at all. 2ℵ20 was a shocking ℵ101, 2ℵ101 was a tame ℵ102, 2ℵ102 was a near-miss ℵ105, and 2ℵ105 was ℵ1946. There was no overarching pattern at all.
It made a kind of sense. Of course the transfinite numbers should be as quirky and individualistic as the finite integers. Why would the behavior of the transfinite cardinals be any simpler than the distribution of the primes, or the set of integer solutions to whole-number equations, or the indices of Turing machines? Why would set theory be simpler than number theory? Why wouldn’t the march of alefs be an inexhaustible source of surprise? There was no simple answer to the Generalized Continuum Problem. Once Jack got past his initial sense of disappointment, the new wisdom filled him with joy.
He didn’t bother telling his insight to Ulla—for one thing they couldn’t talk while in the undersea subdimensions, and for another, she wouldn’t have been that interested. Rather than counting things, she seemed to be studying the curves and colors of the wondrous subdimensional forms.
Jack tensed as they approached the gleaming sheet of the Planck frontier. Would this mean a return to the conniving aktuals of Alefville? With all his heart, he wished to be back home with Ulla. And it was so. The two of them oozed up from the rug on their living room floor.
“I knew the outline of Alefville reminded me of something,” remarked Ulla when they were done exulting. “Alefville is exactly the same shape as this smear of cadmium red that I made on the rug when I was painting your portrait.”
“Alefville is the smear of paint,” exclaimed Jack. “Physical space is absolutely continuous. Every level of the transfinite exists in the small. What a day. And I’m—I’m done with the Generalized Continuum Problem, too.”
“You solved it?”
“Not exactly. But I found a way to let go of it. The thing is—”
“Please don’t start writing a paper about this,” said Ulla. “It is really and truly time for our dive trip. Hey—wait! You already wrote your paper before we left!”
“Sweet,” said Jack, going to fetch the printout. He sat silently on the couch, flipping through the pages. Outdoors the sun was below the horizon and the rain was setting in.
Ulla made a pot of tea. “So?” she said, coming back into the living room.
“It’s all here,” said Jack. “But not in the right form.”
“What do you mean?”
“It’s not an academic paper at all. It’s, um, like a science fiction story. About what happened to us today.”
“But you ran that giant computer search to generate this paper. Did something go wrong?”
“Kind of,” said Jack. “Now that I think about it, there would be infinitely many programs that can write my twenty-six papers and then generate a brand-new twenty-seventh paper entitled ‘Physical Applications of Transfinite Set Theory.’ It looks like I didn’t happen to pick the right kind of program. This is the wrong kind of paper.”
“Oh, so what,” said Ulla, pouring out the tea. “Maybe someone will publish it anyway. Our true-life adventure. Hey, don’t forget to write in that your backache went away.”
Jack penciled in the change and sat up, stretching his arms and smiling. “Done! I feel better already. Let’s head for the South Pacific, Ulla.”
“What about those two aktuals?”
“I think they’re done with us,” said Jack, studying the pencil stub in his hand. Out on the deck sat a toad, enjoying the fitful patter of the rain.
Copyright © 2008 by Rudy Rucker.