Complete Stories (133 page)

“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 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. 2
^ℵ
₄
= ℵ
₅
, 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, 2
^ℵ
₁₀
was a bulky ℵ
₁₃
, although 2
^ℵ
₁₇
was a svelte ℵ
₁₈
.

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 animacule 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. 2
^ℵ
₂₀
was a shocking ℵ
₁₀₁
, 2
^{ℵ
₁₀₁}
was a tame ℵ
₁₀₂
, 2
^{ℵ
₁₀₂}
was a near-miss ℵ
₁₀₅
, and 2
^{ℵ
₁₀₅}
was ℵ
₁₉₄₆
. 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?”

“I’m not sure,” said Jack. “Of course there are 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.’ But my search procedure was supposed to select a junk-free minimal program that takes a very long time to produce its outputs—so you can tell it’s doing some serious computation. But it doesn’t look like I ended up with the right program after all. This is the wrong kind of paper.” Jack paused, thinking. “Or maybe not? Actually, I’ve always wanted to write some science fiction. But I never thought I’d dare to—”

“I’m sure it’s going to be fine like this,” said Ulla, pouring out the tea. “Someone will publish your article 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.

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

Written December, 2008.

Tor.com
, 2008.

In 2008, I heard from the John Templeton Foundation, a foundation devoted to encouraging scientific work that could shed light upon theological issues. They’d recently organized a conference on new concepts of infinity in mathematics, physics, and theology, and they offered to pay me to contribute to their forthcoming volume of proceedings,
New Frontiers in Research on Infinity
, to be edited by Michael Heller and W. Hugh Woodin for the Cambridge University Press.

Back in the 1970s, I’d gotten my Ph.D. in set theory (the mathematical study of infinity), and had published several mathematical papers on the topic. So I was excited about having a chance to publish something about infinity again. But I’d been out of the technical end of the field for so long, that I didn’t want to try and write anything like a math paper. So I thought it would be a good idea to write a science fiction story instead, a tale involving higher levels of infinity and possible physical manifestations of infinity.