Complete Stories (131 page)

Pressing together, Angelo and Rabbiteen crawled into the toad’s mouth. Pushing and pulling, moving as one, the lovers wriggled their way down to the womb of the Black Egg. And of our world they saw no more.

Within the Egg’s twelve-dimensional kalpas, time and space regressed. There was neither room nor duration in which to hunger, to tire, or draw a human breath. Yet in another sense, this was a weightless and limitless utopian paradise in which happy Neetibbar and wry Olegna could gambol and embrace.

The mortal races of the next universe would occasionally comment on two glorious superclusters, titanic arcs of creative energy stenciling the void like a net—sharp and sleek, stable and sweet, weaving the warp and weft of the reborn cosmos.

============

Written October, 2008.

Isaac Asimov’s SF Magazine
, 2009

In April, 2008, I fell under the spell of a popular science book,
The Endless Universe
, by Paul Steinhardt and Neil Turok. I got Bruce to help me work the ideas into this science fiction story, “Colliding Branes.” Bruce had the excellent idea of making the main characters be bloggers, and of having the region between the universe’s two branes be a living being along the lines of a Lovecraftian Great Old One.

As in all our tales, Bruce and I took the transreal expedient of making it a two guys story—and this time out, on of the “guys” is a woman, played by me. Once again we went through ten drafts, with much wrangling via email.

After the last week of the collaboration I went to the hospital with a cerebral hemorrhage—what used to be called a fit of apoplexy. Later I told Bruce that my attack was his fault. He’d pushed me over the edge. Bruce imperturbably replied that I wouldn’t have any further problems if I would just accept that he was always right. You’ve gotta love the guy.

“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. ℵ
₀
, ℵ
₁
, ℵ
₂
, ℵ
₃
, and ℵ
_k
. She also remembered that Jack liked to use his crazy numbers as exponents, like 2
^ℵ
₀
, 2
^ℵ
₁
, 2
^ℵ
₂
, 2
^ℵ
₃
, and 2
^ℵ
_k
. Whatever that meant.

Just as expected, he continued, “In 1873, Georg Cantor proved that for any k, 2
^ℵ
_k
is larger than ℵ
_k
. So 2
^ℵ
_k
might be ℵ
_k+1
, or it might be ℵ
_k+2
or something even bigger. Cantor’s guess was that the transfinite numbers are well-behaved, and that 2
^ℵ
₀
= ℵ
₁
, 2
^ℵ
₁
= ℵ
₂
, and that, in general, 2
^ℵ
_k
= ℵ
_k+1
. I myself think Cantor was a shade too cautious. I think 2
^ℵ
₀
= ℵ
₂
, 2
^ℵ
₁
= ℵ
₃
, and in general, 2
^ℵ
_k
= ℵ
_k+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 make 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 the best 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 typewriter? That takes forever.”

“Not
forever
. Less than alef-null steps. And my little Turing programs are gonna be smart. My search procedure includes an evaluator to weed out the junky ones that spit out garbage, or that run too fast. I’ll find a clean, logically deep Jack Bohn emulator, yeah.” 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 over-excitable 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 hand-write 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. “Bascially, 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.”