Jack and the Aktuals, or, Physical Applications of Transfinite Set Theory

by Rudy Rucker

Overview

Like many other stories and novels by Rudy Rucker, "Jack and the Aktuals" is a wild and wooly dramatization of certain principles of higher mathematics, with added talking animals, sentient pencils, and orders-of-infinity nested within one another like Russian dolls. No description can ever encompass the mind-bending experience of reading a Rudy Rucker story.



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Product Details

ISBN-13:	9781429952835
Publisher:	Tor Publishing Group
Publication date:	07/20/2010
Series:	Tor.Com Original Series
Sold by:	Macmillan
Format:	eBook
Pages:	32
File size:	566 KB

About the Author

Among Rudy Rucker's many novels are the Ware tetralogy (Software, Wetware, Freeware, and Realware); White Light, Spacetime Donuts, Mathematicians in Love, and Postsingular. His nonfiction includes such works as Geometry, Relativity, and the Fourth Dimension and The Lifebox, the Seashell, and the Soul: What Gnarly Computation Taught Me About Ultimate Reality, the Meaning Of Life, and How To Be Happy. He is the great-great-great grandson of the philosopher Georg Wilhelm Friedrich Hegel. Rucker lives in Los Gatos, California.

Rudy Rucker is a writer and a mathematician who worked for twenty years as a Silicon Valley computer science professor. He is regarded as contemporary master of science-fiction, and received the Philip K. Dick award twice. His thirty published books include both novels and non-fiction books. A founder of the cyberpunk school of science-fiction, Rucker also writes SF in a realistic style known as transrealism. His books include Postsingular and Spaceland.

Read an Excerpt

Jack and the Aktuals, or, Physical Applications of Transfinite Set Theory

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By Rudy Rucker, Marcos Chin

Tom Doherty Associates

Copyright © 2008 Rudy Rucker
All rights reserved.
ISBN: 978-1-4299-5283-5

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CHAPTER 1

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. [TEXT NOT REPRODUCIBLE IN ASCII], and [TEXT NOT REPRODUCIBLE IN ASCII]. She also remembered that Jack liked to use his crazy numbers as exponents, like [TEXT NOT REPRODUCIBLE IN ASCII], and [TEXT NOT REPRODUCIBLE IN ASCII]. Whatever that meant.

Just as expected, he continued, "In 1873, Georg Cantor proved that for any k, [TEXT NOT REPRODUCIBLE IN ASCII] is larger than [TEXT NOT REPRODUCIBLE IN ASCII]. So [TEXT NOT REPRODUCIBLE IN ASCII] might be [TEXT NOT REPRODUCIBLE IN ASCII], or it might be [TEXT NOT REPRODUCIBLE IN ASCII] or something even bigger. Cantor's guess was that the transfinite numbers are well-behaved, and that [TEXT NOT REPRODUCIBLE IN ASCII], and that, in general, [TEXT NOT REPRODUCIBLE IN ASCII] I myself think Cantor was a shade too cautious. I think [TEXT NOT REPRODUCIBLE IN ASCII], and in general, [TEXT NOT REPRODUCIBLE IN ASCII].

"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: [TEXT NOT REPRODUCIBLE IN ASCII].

(Continues...)

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Excerpted from Jack and the Aktuals, or, Physical Applications of Transfinite Set Theory by Rudy Rucker, Marcos Chin. Copyright © 2008 Rudy Rucker. Excerpted by permission of Tom Doherty Associates.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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