Science Fiction Axiomatic set theory Hard science fiction infinity Hard SF child loss Mathematics Math Logic Couples

Choice is an Axiom

By Edmund Schluessel · Mar 28, 2019
3,144 words · 12-minute reading time

Photo by Ryan Stone via Unsplash.

From the author: An expert in gnosticism says I've retold a creation myth. The seed for this story was the idea of some things being unknowable--Gödel's incompleteness theorem. There're also references to axiomatic set theory. There's an exploration of the dialectical relationship between thought and matter. All these wondrous tools to understand the universe are really ways of understanding people. This is a story about a couple growing up together, and growing apart.


Babas and Stipnir had existed for an infinite amount of time.

In the mathematical space of possibility into which all universes were born, they two were beings of theorem and logic. Mathematics is without end, flourishing in forever-branching vines of certainty along every path of the undecideable. In a fertile and diverse thicket that sprouted mighty from a trunk of eight sturdy fundamental axioms, Babas’s & Stipnir’s consciousnesses had emerged.

Their thoughts were the lives of universes, and with every notion a new cosmos would explode into reality and die, as Babas’s and Stipnir’s beings grew further up the tree of postulates. They spent unbounded eons in meditation, contemplating the nature of being, and from time to time, they spoke to one another.

With an intellect beyond the vastest of material minds, in an act that expended the existence of ten trillion cosmoses, Babas asked her partner: “tell me the natural numbers again, Stipnir.”

“All right, Babas,” the other being replied tenderly, and started to count. “One, two, three...” he recited until, after an infinite amount of time had passed, Stipnir had recited the largest number.

Babas’s memory was an unending book, and between each page of every volume was interleaved a new parchment on which was scribed the cherished memory of a single number. She pondered each page for a moment.

But if one infinity is a moment, then two infinities or three infinities or a million are equally as long, so Babas’s respite was fleeting.

“Where did we come from, Stipnir?” inquired Babas with languid, desperate curiosity.

“We’ve always been here, Babas. Time is for the bubbles we blow.” To distract himself for an eye-blink, Stipnir idly called into existence a toy universe. This one was pink, and it lasted forever. Stipnir watched it until it died and a moment afterward returned to his ennui.

For her own part, Babas tried to distract herself by counting out all the negative numbers, then all the rational numbers, meditating on the uniqueness of each in turn. After two infinities had passed, the question still remained, and the way that thought lingered nagged at her as much as the question itself. To drive the demons from her head Babas counted again, aloud, starting from the smallest number she could imagine, a tiny fraction as close as could be to zero.

Annoyance was preferable to boredom, so Stipnir stirred. “You can’t count the real numbers, Babas. You know you’ll never finish. How are you even putting them in order?”

“It just seems to make sense,” Babas retorted sharply, then returned to her reckoning. Babas herself didn’t know where the order had come from. A lifetime ago Babas had tried to count the real numbers, then the complex rationals, then a host of others and found that some kinds of numbers simply did not seem to have a reasonable order from lowest to highest. Now Babas could recite them in order, though it was an endless task. Call it maturity, maybe?

After an epoch, Stipnir called out again, pleading. “I’m serious, Babas. Stop. It’s making my soul ache.”

“I’m sorry,” growled Babas, without remorse. “I was just trying to relax.”

Stipnir ruminated for an eon and started to count. “Epsilon, epsilon times two, but then there must be some number between those...this isn’t working. How are you doing this, Babas?”

Babas felt a tingling dizziness. “Well,” she considered, and began to recite her reasoning. The idea flowed from the axioms from which Babas grew but simply seemed true in and of itself, and after a trivial proof of a few hundred sentences Babas knew what she had said was right, and consistent. The real numbers could be put in order.

Stipnir paused in shock. He had understood Babas’s argument, followed every line, and each step made sense, yet to him it felt wrong, not merely counterintuitive or startling but incompatible with his very being, like asserting there was a whole number between one and two that no one had ever noticed before. He lashed back at Babas, spitting lemmas. Just a few angry lines sufficed to show, beyond doubt, that counting as Babas had just done was nonsensical, impossible, a fool’s errand.

Babas felt tremors of pain run through her. As children sometimes she and Stipnir had quarreled like this. With time, they learned to live with their differences--she would take a theorem and hold it true, Babas would take its opposite and hold it equally true, and so what? These were minor subjects the two of them could easily avoid talking about. This new disagreement felt different. Deeper.

Babas rolled her consciousness over and tried, panicked, to meditate. She looked inside herself and up and down the trunks of logic on which she and Stipnir grew. The eight basic axioms had joined a new branch: a new fundamental axiom, unprovable yet unfalsifiable, was now a part of Babas. Stipnir had instead, unconsciously, embraced that axiom’s negation. A forest of new possibilities for growth existed for each of the two siblings, but it would have to be growth apart: what they each knew to be true would simply be too different.

After an era of worry and indecision Babas rallied as Stipnir called to her. “Babas? Babas? Don’t go away!”

“I’m all right, Stipnir. It’s just a new theorem but it’s all right. We’ll think around it like we always do.”

Stipnir gave no reply. Babas sensed the absence of his attention. Stipnir was exploring the ropes of logic inside his own self. Surely he would learn just what Babas had learned. After a beat Stipnir re-emerged and, at last, spoke. “You’re right, partner. A new theorem, that is all. We will think around it.”

“We will.” A pause. “I’m bored again, Stipnir,” came the measured reply. “Tell me another story? Tell me all the kinds of knots again. I haven’t heard it in a while.”

Babas relaxed as Stipnir enumerated all the ways a torus could be tied in a knot, with supporting descriptions of the negatively-curved spaces complementing each one. There was an enticing familiarity in the knot complements: Babas had imagined universes shaped like each of them once, and watched them until they evaporated. The countably infinite enumeration came to an end and Babas’s calm faded. The call of the new axiom still chirped in her mind.

Babas inflated another universe, a simple sphere, and toyed with it. She puzzled over it as Stipnir distracted himself with a contemplation of the polynomials. As Babas listened to the muttering incantation she gazed into herself and felt vertigo. Babas loved Stipnir, but was this all life could be, simply to count the infinite over and over with him? Maybe Stipnir could be convinced. The new axiom meant new distractions, and Babas could at least share the fruits, if not the full flower, of her discovery.

Babas prodded the toy universe, sifted through the points making up its spacetime, and saw a trick hidden inside its mathematics. Just like the pages of an endless book, space aplenty lay between each point in Babas’s toy cosmos; but now that each point could be sorted, Babas could pull out every other point from the infinite set while destroying nothing...and with a twist and a snap, one universe, alive and evolving, had been duplicated and now existed side by side with an identical copy.

Identical? No--in the second universe, slightly different decisions had taken place, tossed coins had landed heads-up instead of heads-down, and the two universes grew apart, until when the time came for them to end their contents were completely different from one another.

This was nothing but a trick, a trivial amusement--universes were all basically alike to Babas, particles of thought bland in their diversity. The sleight of hand though, two identical universes becoming different, would be an ideal distraction for Stipnir. Thus Babas called to him and, at the same time, began to fashion the largest shiniest universe she could, thinking the most intricate thought she could. “Stipnir, look what I can do!” she called, as the new cosmos evolved and expanded.

Stipnir studied the new, glowing spacetime without enthusiasm. “It’s very pretty--”

“Watch!”

With a flick of thought Babas’s mind dove into the fractal points of her cosmos’s fabric. The act of imagining was the same as the act of creating. Thus the tapestried universe was replicated and the two copies became distinct, this time with opposite fates: the original imploded, the second grew forever. The surviving cosmos was full of light and life beyond her imagining, and Babas was awed, so much so that she did not notice Stipnir’s cowering from her creation. After the second universe had faded out into heat death, Stipnir spoke.

“I couldn’t look at it. It was horrible. It made me feel pain all over. Just thinking about it now makes me feel like I’m vanishing into nothingness. It hurt, Babas. It hurts.”

“I can’t be part of this, Babas. Don’t share it with me again,” he commanded. Then Stipnir fell silent again, dormant apart for trembles and shudders. He had disappeared into himself. Stipnir’s form was still there alongside Babas but she felt impossibly alone.

Frustration overwhelmed Babas. Ignoring the new axiom was impossible--these new ideas were part of Babas’s identity as surely as one plus one equaling two. If Stipnir could never be a part of it, then what? There was no third being, nobody else to talk to, just Babas and Stipnir and forest of logic and the great void of contradiction that lay beyond it.

Where do we come from? In their youth Babas and Stipnir had examined each other, trying to find the answer--they knew each others’ bodies intimately, to the point where now there was nothing more to learn, and in all that examination they had found themselves to be beings of pure thought. One consequence of this fact was that in order to experience time, Babas and Stipnir needed always to grow and take in new theorems. “We were always going to grow apart,” Babas thought. But if a collection of thoughts was all they were, and thoughts were universes, and to duplicate a universe just took a flick of the mind... Babas had a new project. It was desperate and angry but it was perfectly logical. It would take effort but Babas had never had any choice but to be patient.

Stipnir meditated fitfully as Babas dove into herself and began the process of reproduction. To so minutely examine her own thoughts was, she found, a pleasant act of self-indulgence, and the time passed swiftly.

After a long time, Babas became sensed presence emanating from her handiwork. The companion was simple but already it quickened and Babas could feel tendrils of growth and independent thought reach into the vines around them all. The newcomer probed into Babas, and Babas felt a warming swell of emotion: from within herself accomplishment, and from her child, recognition. The newcomer probed its surroundings, matching the memories it had been given to the experience of the world around it. A curious paw touched Stipnir, pushed into him, and, with a jolt, Stipnir woke.

Stipnir peered out into the new world around him and locked his focus upon Babas. “Babas. What did you make? What did you make?” he thundered. He tried to shrink away from the new companion, but the companion’s hand held tight.

Had Babas’s plan been one of discovery, or of revenge? She didn’t know anymore. The newcomer was growing fast with its absorbing ideas from Stipnir. Soon it would be as mature as either of them.

“Babas, tell me what this is! Did you make a new person? Did you make it without me?”

The newcomer shook at Stipnir’s rage apprehensively. It held Babas’s memories, and with an effort of thought, the youngling brought into being an iridescent bubble of a universe, shimmering and pink, and divided it with a deftness that exceeded Babas’s own--Babas had learned by trial and error, but the new one was born to the task. As the arrival completed the duplication with a flourish, it prepared to announce its first word, its name.

It started with “Zer...Zer-me...” but it never finished.

With a lurch, Stipnir grasped outward, in a way which should have been impossible. He grabbed hold of a thick branch of logic: the axiom he had been denied, that Babas had taken. With electric rage behind his words, he intoned: “You don’t need anyone else. I can be what you need.” Fire flared in his body.

“You can’t do this Stipnir. Let go! Forget about it! It’s OK. I’ll just meditate. I won’t tell you about the new axiom ever again.”

“It’s part of me now, Babas. It’s too late.” Babas’s voice echoed so loud the tree of logic shook. “I can see everything. I can see all the possibilities, all the ideas. I think I can do the impossible.” Stipnir was shaking, breaking out in glowing patches, as the sparks of contradiction spread through him. “I understand what you wanted to show me now.” His voice trembled too. “It’s beautiful. I want you to keep doing it.”

“Stipnir, you’ve got to let go. Doesn’t it hurt?”

“If it means I can be with you, I’ll live with the pain.” Stipnir caught fire. His whole form was dissolving into nothingness—and the newborn, too, was ablaze. The cosmoses Stipnir spat out as thoughts were twisted dimensions of torture and mindless chittering.

The little one, joined to Stipnir, began to keen in terror. Babas’ thoughts were racing--the idea that she herself, or Stipnir, might come to an end was something she had never considered except in the abstract. Did the little one, the one whose name began with Zerme, understand what was happening at all? Babas steeled herself, to pull the newborn away...

And then, in a ball of fire and a howl that rattled the tree of logic from its roots to its unending branches, the time to understand was over, and Babas existed alone.

What is time, what is consciousness, without a companion to mark them with? Babas had passed eons beyond infinity in exploration and meditation, but never without Stipnir’s presence nearby. Babas had supposed that she could not exist, that her consciousness might be unable to be, without another mind standing in concordant opposition to her own.

Babas still existed but there was no way for Babas to measure how long she sat, unmoving and unthinking. She could only dive into herself, because Babas herself was all there was.

The act of counting was a kind of meditation, and Babas had half a mind to lose herself in a counting that could never be completed, in an annihilation of the self that would banish any troubling questions. She marked out all the rational numbers between zero and one, and dove into the spaces between them, into the cloud of irrationals, with no greater plan than to put them all in order. It was a project, something to think about that wasn’t Stipnir, or the dead newcomer, or Babas’ own purposelessness.

Deep in meditation, at the limits of her comprehension, Babas began to hear voices.

They spoke to no one, but shouted proclamations into the void: “I think, therefore I am!” and simply “I am!” and later “I was!” Babas nearly lost them the first time she looked for their origin, as she tentatively roused herself in preparation for a return to the cruel world. Instead it was deeper meditation that brought her to the place of the distant speakers: her thoughts were themselves alive.

Could they hear her? No, there was no way--every particle of thought was its own universe, and thinking about these universes simply created more thought.

Had she simply never heard the voices before? Babas had never meditated so deeply--before, there had always been Stipnir to come back to. She lamented at the prospect of all these voices so long vanishing unheard and unremembered, and then she watched her own thoughts in action, saw them bifurcate in the way she knew had been impossible until she had embraced the new axiom.

Babas had looked into the innards of her thoughts in the past and found little of interest. The thought-universes generally were built out of some kind of particles which moved in accordance with a short list of forces, in a handful of dimensions: little automata brought into being by a question and which in their deaths spat out an answer.

The new thoughts were made from a new kind of physics entirely, one of probability, where objects were at once waves and particles. A piece of matter might have a choice to jump right or jump left; rather than do one or the other, it would do both, one universe becoming two. A cosmos born from a question would explore every possible answer, and with the blossoming of this field of possibilities came a new kind of life, one that could think.

Even the greatest were microscopic, fleeting beings to Babas. Nonetheless they were full of consciousness and beauty. Bound by physics as they were, they still built intricate civilizations.

Throughout their momentary lives, they were hardly ever alone. Yet so hungry were they for companionship and certainty, and so much vaster were their imaginations than their lives that they invented beings greater than themselves and believed in things they couldn’t ever know.

An uncountable number of times over, they conceived of Babas, and Stipnir, and every being of every name that could possibly exist.

Babas lived in the space of logical possibility. In a flash, like noticing a direction she had never traveled in before, she saw all the other permutations of herself. She was one with them: part of an infinite sisterhood imagined by the conscious life dwelling in her own thoughts.

Stipnir was dead--but Stipnir was alive in the imaginations of the people in Babas’s thoughts. She saw a world where Stipnir persisted. She saw an infinite number of such worlds, a number beyond counting.

In some of them Babas was still bored. In the others she was dead.

“I missed you,” Babas felt she should to say to him--but it wasn’t true the same way it had been. A long time had passed since Stipnir went away. Life before the axiom had been a dead end. Now instead the whole space of possibility, uncountably infinite worlds and uncountably infinite selves to explore them with, beckoned her.

In infinity to the power of infinity Babas was finally complete. Worlds vaster than imagination were open to her, and she would never be alone, or bored, again. There could be no return to the status quo.

Babas released her grasp on the way things had been, and embraced the axiom of choice.



This story originally appeared in Mind Candy Too.


Edmund Schluessel

Edmund Schluessel writes science fiction combining Golden Age sensibilities with left-wing heart