by Various
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:
“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 And, as it happens, we also proved that as well. And for the next three levels, things flatten out. That is, we proved that and
“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 If I had to bet, I’d take a wild guess that for any infinite cardinal after we have more or less like Georg Cantor expected. There’s only that one anomalous double step at the start, where shoots up to
“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 to
ngue, 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.