Transfinitude by CB Anderson


Luis was preparing a paper. It was February, cold in the evenings when he walked home through thin air to the Central Square condo he shared with his wife and infant son. In June he would present his work at the American Mathematics Society in Baltimore and, if all went well, he’d be up for promotion to associate professor in the fall. The abstract had been sent ahead. The writing of the paper should have preceded the abstract but six-month-old Theo, while imagined in some hypothetical future, had not been planned and things were a little off course.

The topic of the paper was Cantor’s continuum hypothesis. Day by day, Luis was trying to forward his argument for a non-local approach to the establishment of axioms from which to prove or disprove the postulate. “We cannot rest on the dubious notion that CH is simply unresolvable,” he’d stated in the abstract, although he did not possess the same urgency he had a year ago, before Sarah learned she was pregnant. Nor was he progressing in his own work on supercompact cardinals, which was essential to the case he was making in the paper.

The baby was beautiful. Locking his office door and descending the stairs to the lobby, Luis conjured Theo’s face: gray eyes, auburn curls, heart-shaped mouth. Baby smell of milk and diapers, one that warmed Luis but could also bring on a faint despair. He hurried on the frozen sidewalk. Snow was walled three feet high on either side, a tunnel from home to work and back. Sarah would be upset if he was late. She’d taken leave from the university genome lab to care for Theo, but she did not seem happy. Luis could do little without correction: load the dishwasher, bathe or feed the baby. That morning she’d chided him for failing to awaken during the night with Theo, to change and bring him to her to nurse.

“Just wake me when you hear him,” he’d told her as he spooned coffee into the Krups.

“I really shouldn’t have to.”

Luis broke eggs, while in Sarah’s lap the baby studied his toes.

“The whites are dripping on the counter,” she said.

He was home—a teal Victorian with a mansard roof under which sat their four third-story rooms. Luis loved the condo, old and detailed in a way that little in the Tampa suburb where he’d grown up had been. But looking up at the ornate windows, he realized that, distracted, he’d forgotten the marinara Sarah had texted him to get. And the mint chip ice cream. He deliberated—backtrack to the Superette or go inside on time—then opened the door.

Upstairs, the air was steamy from the pasta Sarah was boiling. Theo was with her in the kitchen, batting at the rings suspended over his bouncer. He squealed as Luis bent to kiss him on the forehead, bounced harder.

“I’m sorry, I forgot to get the sauce,” Luis said. “I can go back for it.”

For a few moments she remained with her back to him. When she turned, her eyes were wells of displeasure.

“I’ll go back.”

“It’s too late. We’ll have to eat it plain.”

They were quiet at the table, pent-up. Theo, still in his bouncer, began to fuss. Luis chewed pasta—sticky, bland, his fault. “How was your day?” he asked.

“Alright. We went to PlaySpace and bought onesies on the way home.” A pause. “How was yours?”

Luis understood: Sarah being home meant his work should flourish. “Pretty good,” he said. The topic of his paper would be quicksand. “You remember my student, Mike Liu? He got a post-doc at Michigan. And Martin wants me to teach set theory next fall.” The clock on the kitchen wall ticked out seconds; the one in the living room answered. “So things were okay today?”

She sighed. “I’m tired, Luis. And when I’m not tired, a lot of the time I’m bored.”

Luis glanced at Theo, worried he’d somehow understand, but the baby was leaning from his bouncer to inspect the spoon on the floor.

Abruptly he stood, felt a physical shrinking inside as he carried the dishes to the dishwasher. When he’d loaded them, he found Sarah curled in their bed, nursing Theo. Seeing them, mother and baby look-alikes nestled in the comforter, chastened him. His eyes stung. He had to try harder. Be more understanding. He resolved to wake that night when Theo did, to sleep so lightly that he’d hear him even before he cried. He put on his coat. “Going out for that ice cream,” he told Sarah. “Back in ten minutes.” He bought mint chip and two other kinds, but when he got home Sarah was asleep. Theo lay beside her, awake, kicking his little legs. Fear stirred in Luis. The baby could have rolled off the bed. He could have gotten caught in the covers.

Theo was fine. Luis carried him into the living room. It was snowing, large showy flakes that wouldn’t amount to much. Light from the streetlamp captured the descent. Theo leaned to watch. “Snow,” Luis told him, “nieve,” pressing his cheek to the child’s soft hair. He loved winter, when white wasn’t just a color but a way of being—orderly, internal.

All the snowflakes that ever had fallen and would fall, infinite yet countable. Aleph-naught cardinality, the same as the integers. Luis had spent the afternoon pondering the givens and now, again, CH was there, niggling. The hypothesis held that there existed no set whose cardinality was between that of the integers and that of the real numbers. But what if Cantor’s suppositions were faulty? Wasn’t an endlessly extended line less than the plane upon which it lay?

Theo had fallen asleep, his breath soft and slightly sour. These were good moments—the house quiet, the day’s pressures fallen away, Sarah’s face relaxed in sleep. Luis carried the baby to his crib, then changed his mind and went back into the bedroom. He pulled out the portable carrier, a clumsy palm-woven affair they’d gotten from his aunt and never really used.

Outside the snow already was letting up. Frost pearled the window. Louis set the carrier on his side of the bed, along the wall. He laid Theo on his back and covered him. Sarah might not approve, but it was worth it. This way he could watch them both.


Growing up he’d felt alone. His classmates at Tampa Prep had skateboarded and surfed the Gulf when they weren’t in school but Luis—though athletic, a soccer player through middle school—spent much of his time online. By his junior year he’d grown tired of Warcraft and Diablo, found himself drawn to philosophy chat rooms where he soaked up Plato and Khayyam, later Russell and Frege.

Discovering Galois, he worked through permutation groups on his own. He hadn’t particularly liked math when it was about finding the right answer, but proofs were stories told with symbols. The melding of intuition and logic to render a universe freed his imagination. When on a Sunday his mother would finally cajole him outside into green grass and sun, it seemed to Luis that he had no place in the bright suburbanscape of Palm Garden. He mowed the lawn and skimmed the pool with his father, but it was as if his real life waited for him somewhere else.

He found it in Cambridge. Before classes started, he stood one afternoon in Harvard Yard with his new roommate, Evan, a physics major from Rhode Island. It was early September, the day cold enough for a sweatshirt. Squirrels leapt from tree to leafy tree, and the T rumbled underfoot. “Let’s score some burgers at Bartley’s,” Evan said. Sitting in the vinyl booth with two chilidogs and onion rings, Luis felt the vectors of his life sliding into alignment. He’d miss his family, especially his mother, who loved him even if she didn’t understand him, but visits home would be enough.

At 28, in grad school at MIT, he met Sarah, a genetics post-doc already known for her work in sequencing. On an early date they raced their bicycles through the nighttime streets near the university. As Luis pulled alongside her, she threw back her head and laughed. Streetlight shone on her face, and he knew he would love her. Within months they were spending almost all their time away from work together, hosting elaborate dinner parties during which they served sangria made from the most unusual fruits they could find at Star Market: rambutan and ugli, or pitaya. After dinner they gathered in Luis’s living room to play Go or to listen to Evan’s girlfriend on her fiddle. To Luis, so much happiness did not seem possible.

His mother, visiting the winter before he received his PhD, remarked that he seemed settled in himself, solid, and Luis could feel it—a new certitude of stance. He had Sarah and his friends, and he had math. His papers were well received and one, “Expanded Ternary Relations: Implications for Zermelo-Fraenkel and the Axiom of Choice,” was important enough that the Ivies started courting him for tenure track. He knew, even as it was happening, that he was less brilliant than thorough and persistent, and lucky.

So he was back in Harvard Yard, thirty-one months into a three-year appointment. The warren of junior faculty offices was an active, busy place that until recently had enlivened him. But in the last month he’d begun taping paper over the mesh glass of his door on days he didn’t want to be disturbed. He hoped to suggest serious work in progress, though the opposite was true. On those days, Luis lay on his back on the antique Saroukh rug that Sarah had surprised him with when he moved into his office. Adrift, he sensed the surrounding buzz of his colleagues’ cognition, a tangible force that threatened to consume him.

It was mid-April, and still he hadn’t made much headway on his proof, let alone gained momentum in the overall argument of his paper. It was known that the least strongly compact cardinal need not be supercompact. Luis was trying to argue that the consistency strength of the two large cardinal notions was nonetheless the same. Daily he waited for the moment when everything would shift and understanding would arrive, the way it always had before. But nothing came, and too often the afternoon was a slough of tedium and a rising sense of unease.

He looked forward to the days when Sarah had an appointment and left Theo with him. Luis would walk with the baby around the office, play with him on the rug. As he diapered or fed him, the background mental hum finally would recede. That afternoon Sarah had dropped off Theo on her way to the hair salon. Luis got up off the floor to let them in. Now he watched the baby, asleep in his stroller, eyes scrolling behind their lids as he worked through an infant drama. Luis drew a breath, looked back at the screen of his laptop. The cursor ticked, impatient. No new work to save today. Instead his mind again reached backward, maddeningly. What if there wasn’t a one-to-one correspondence between the natural numbers and all countable sets? Then proving CH would be moot.

The idea was farfetched, regressive, but Luis succumbed to it. He was deep into imagining petals looped from square numbers, where 25 had 10—the number of non-inclusive integers between it and 36—and 36 had 12, on and on: 2 x, where x was the root of the smaller square. Eventually the petals would approximate a solid band, ever denser as the total approached infinity. How could such a sequence be counted in the same way as the squares themselves?

At 3 p.m. Theo opened his mouth in the protest he made when he crossed back from sleep to wakefulness. Luis had learned to catch it early, before the complaints turned to full-blown cries. He lifted the baby from his stroller and jiggled him in appeasement. Theo offered a tentative smile, his breath rising to fog Luis’s glasses.

In the faculty lounge Luis warmed the bottle Sarah had packed. He sat by the window and offered the milk to Theo, who placed his hands over Luis’s as he drank.

“Gosh, he’s really getting big.” Cecily, a geometer and Murray Prize finalist, took a seat across from them. “What is he now, eight months?

“Thirty-two and a half weeks. Thirty-three on Sunday.”

Cecily paused, as if Luis’s knowing Theo’s exact age was odd. “He’s very cute.”

“Thank you.” But Luis thought Theo was more than cute. The curved eyebrows, whorled ears, the starfish hands over his own—they seemed proof of a universal rightness. He looked down, locked his gaze into the baby’s.

“—topology seminar next month,” Cecily was saying. “Hope you can make it.”

Theo finished the bottle with a smack, opened his arms wide in a what’s-next gesture. Luis wiped milk from the baby’s chin, stood up from the table. “I’d better grade my Calculus exams,” he said.

Back in his office he settled Theo on the floor, pulled out a clean Huggies and unfastened the old one, which he wadded to toss into the trash. Two points. As Theo thrashed and rolled onto his belly, Luis taped on the new diaper. Without Sarah’s oversight he felt his own competence.

As easily as he’d rolled, Theo pushed up onto his knees, then over onto his bottom. Grinning, he sat. All this in the last two months. Luis pulled a pop-up toy from his file cabinet. Theo squealed, banged on the colored bars as Luis watched. The exams had been an excuse; they were already graded. He relaxed into Theo’s play, laughed with him when a green cow popped up.

The knock at the door surprised them both. Theo’s mouth formed an “O” of alarm. By the time Luis got up, there was a second, more insistent round. Annoyed, he swung open the door then scrambled to rearrange his expression into one of welcome.

“Martin. Come in.” His department chairman stood inches away.

“No, no. I have to sit for Jay’s defense. Again.” Luis felt a wash of sympathy for Jay Miller, whose dissertation had failed the first time. “One of your students was looking for you.”

“Really, who?” Luis was surprised. He’d been here since his morning class. Had he dozed on the floor?

“Young woman from your Calculus class. Dinah, Dina?”

“Diana Metzger. I’ll email her.”

Martin poked his head into Luis’s office, took in the general disarray and Theo peering up at him. “How are things going?”

Luis knew Martin meant work, his paper—he alone would represent the university in Baltimore—but a sudden perverseness took hold. “He’s doing great, thanks,” he said. “He’s sitting. And he has two teeth, two more on the way.”

Martin nodded. “Well, good.” He waited.

The lie came easily. “I’ve shown a couple of corollaries.”

“Wonderful. Which ones?”

In their field, a fact wasn’t a fact until proven, and Luis was good at weaving the deductive threads. His craftsmanship had landed him this position. But he hadn’t budged in showing that the consistency strength of his two large cardinal notions was the same. “Martin,” he said, gesturing to Theo, who’d hitched on his bottom to where Luis stood and was sitting on his shoe. “I wonder if now—”

Both men turned: Sarah, back from her appointment. Her auburn hair was cut below her chin and blown smooth. She looked beautiful, happier than Luis had seen her in weeks. He hugged her, “Great haircut,” inhaled the scent of salon products. He felt a rush of emotion, remembered all the times she’d come for him after work and they’d biked home together, stopping for burritos or pho along the way. There’d been no lack of conversation then.

“Good to see you.” Martin was beaming. He liked Sarah, became courtly and avuncular around her.

“Thanks, Martin. It’s nice to be out.”

“The little one looks to be thriving.”

“He is. He’s almost eight months now.”

Thirty-two and a half weeks, thought Luis.

“Any thoughts on when you’ll go back to the lab?”

When she bent to lift Theo, Sarah’s hair fanned gold. “Actually, I stopped there on my way here. I’m going to start back soon.” She turned toward Luis. “Three afternoons a week.”

Why was she telling him here, now? He reached for something to say.

In the silence Martin backed away. “Better keep moving,” he said. “Can’t say I’m looking forward to Jay’s defense.”

Sarah and Luis moved into his office. “Poor Jay,” said Sarah. “I hope he makes it through this time.”

“I hope so too,” said Luis, “but Sarah, why didn’t you mention that you were thinking about going back?”

“It’ll be fine.” Her tone was matter-of-fact. “The Center has an opening for an infant. I called this morning to double-check. And it’s my decision, isn’t it?” She ran the hand that wasn’t holding Theo along a bookcase. “Maybe you should dust in here.”

Luis’s chest felt tight. “Maybe you should stop telling me what to do.”

Sarah shook her head, opened her mouth then closed it. Luis understood: they would argue later. Wordlessly she fitted Theo into his stroller, picked up his bag and left. Luis sat at his desk and powered down his laptop. Below in Harvard Yard, undergraduates hurried from the library to their dorms. The day had been warm, the air heavy with snowmelt, but the ground was still frozen and the night would be cold. The sky was leaden— Luis felt as if it were coming through the window to settle around him. He didn’t move. In Palm Garden, the sun had shone and shone, one day indistinguishable from the next. After fourteen years in Cambridge, he felt first relaxed, then dulled, and finally exhausted whenever he returned to the Florida heat. Always he was glad to get back to New England, to be pushed along by wind or enshrouded in fog.

The door to the seminar room at the end of the hall opened and closed. Luis heard voices, wondered how Jay’s defense had gone. His own dissertation had been on fractals and generalized Cantor expressions; for two years he’d pored over Cantor’s extensive works. He knew the lore: that Cantor, a devout Lutheran, had claimed his theories on transfinity—infinities beyond infinity—were communicated to him by God; that some considered his work brilliant while others dismissed it as outlandish; that Cantor grew despondent and was hospitalized repeatedly for depression, that he came to question his religious beliefs. But last year, in a footnote to a journal article, Luis learned that Cantor’s son had died suddenly at age twelve. Reason enough to doubt God.

The light was almost gone, the figures below reduced to motive objects. Luis switched on his lamp. His beliefs were cautiously agnostic; he and Sarah had both been raised Catholic; their distance from it, their choice not to baptize Theo, had made him feel closer to her. They were, he thought, very much alike in the ways that mattered. He leaned back, rubbed his temples. He shouldn’t have snapped at her. Sarah was right; it was her decision whether and when to return to work. Their marital transaction allowed for this: Sarah’s single-mindedness in return for her clarity and drive. His own place was less certain.


The baby had been sick with a fever and cough. The pediatrician at University Health Services ruled out croup, bronchitis and ear infection. “A minor respiratory virus,” she’d told Luis and Sarah when they brought him in, but it didn’t seem minor to Luis, not with a fever of 101. Whatever it was, he was sure Theo had contracted it at the Center, and it was only his fourth week. When Luis picked him up on the afternoons that Sarah was at work, he was surprised each time anew by the ubiquity of babies: in swings and high chairs, on foam mats, playing, crying, smiling, sleeping. Sneezing. He’d seen a little girl sneeze on Theo, and then he got sick.

Today, a Friday, Theo was better. His fever was down, his forehead cool when Luis pressed his lips to it. He kicked and babbled, laughed as if delighted to feel well again. He hadn’t been outside since Tuesday, the past forty-eight hours a black hole of sleeplessness, Tylenol, and online symptom searches. Luis considered bundling him up and taking him to Cecily’s topology talk. It was warm enough—nearly 60—and there were hours before his afternoon nap.

Five minutes later they were on their way, a good move—they both needed fresh air, and not showing up at the seminar would be less than politic. Cecily, who’d been at Harvard only two years, had nudged out Luis for the Murray Prize nomination; people would read into it if he didn’t go. He decided not to text Sarah for approval. Luis didn’t blame her for the baby’s illness—going back to the lab had been right for her; she came home happy on workdays—but being alone with Theo meant he made the decisions.

Outside on the Green Street sidewalk, the stroller rolled along. Spring-blue sky, the bare branches overhead alive with birds. A woman chipped vestiges of ice from her driveway with a spade. Theo rode the way he always did, semi-reclined with his feet propped up on the crossbar. Passersby smiled, first at the baby then at his father. Luis enjoyed interacting with the world with Theo as intermediary. He nodded and smiled in return, his mind pleasantly emptied.

At Hancock Street, where he should have gone right to take the back way to the university, he found himself turning left, down to Kinnaird then right on River all the way to the Charles. There was just enough time to show Theo the river, which was high in its banks and turgid when they reached it. Theo leaned forward to survey it. “River,” Luis told him, then “ducks” of the Mallards on the bank. He was reminded of last summer, a perfect August afternoon and Sarah very pregnant as they walked through Boston Garden. On the small pond there, ducklings had chased their mother—Sarah laughing when one of them paddled so hard he propelled himself atop the water. Theo had been born the next day.

It was warm—Luis could almost feel the heat of the sun on the baby’s face. “Nice, isn’t it?” he said. The water lapped the bank, glittered as a breeze came up. Luis rotated his shoulders in their sockets and inhaled deeply. The miasma of fatigue that had descended as he sat up nights when Theo was sick was lifting. As more ducks flew in and skidded to a stop on the water, it came to him: The key to his proof was the construction of a canonical inner model, one for supercompact cardinals. If he could establish iterability for the inner model, the rest would follow.

It was the shift he’d been waiting for, the wide-angle view that gave him start and end points. To join them would take weeks, maybe months, but finally he saw the argument. It should have sent him rushing for his office, to redirect, to open a new file and begin. Instead he got up, released the brake on the stroller and retraced his steps: River to Kinnaird to Hancock, but this time left on Green to Harvard Square. They would ride the subway to the Garden and check to see whether the ducks were there this year. Tomorrow he’d tell Martin what he now understood about the proof.

Underground he bought a packet of caramelized cashews and pulled out a bottle for Theo. As they waited for the train, he began to make notes on his phone for the new proof: Establish iterability by induction on models of determinacy using supercompactness measures over . He stopped, removed the blanket from around his son so that he wouldn’t overheat. The immediacy of that act, the feel of the small firm torso beneath his hands, set off a chain of images: Theo on a bike as Luis ran alongside; on a soccer field with other kids; on skis for the first time.

When the train arrived, Luis sat across from a teenaged couple, entwined, who swayed in tandem as the train lurched out of the station. Luis gripped the stroller. The walls of the tunnel flashed occasional graffiti before they went black. The teenagers were dressed alike in jeans and hoodies. Luis imagined their uncomplicated lives: school, sports, sex, sleep, friends, except that they of course did not consider their lives uncomplicated.

When Cantor’s son had been dead four years, the Royal Society of London awarded him the Sylvester medal. That same year, at the Congress of Mathematicians, a Hungarian colleague presented a paper that ostensibly disproved the tenets of Cantor’s transfinite set theory. The next day, another colleague showed fallacies in the Hungarian’s argument—but Cantor’s daughters, whom he’d invited to attend the Congress to hear his work, already had returned home.

Momentarily, the teenagers broke apart. The girl made eye contact with him then smiled down at Theo, who wide-eyed watched her back. As the train sped through the darkness, he leaned back with his bottle and propped his feet. Love washed over Luis. The train slowed, stopped, started again. Somewhere between Central and Kendall, he understood he likely would not go to Baltimore, might or might not seek the promotion to associate professor. This surprised but did not upset him. He might or might not finish the proof.

At Park Street, he stood and wheeled Theo onto the platform. It was crowded, trains running continuously. A trio of musicians played something classical yet discordant. Luis tried to go around but as he passed them Theo began to cry. It was 2:15. The topology seminar had started. At her lab, Sarah was loading the afternoon samples into the thermocycler. The cries built then overtook the music. People walked around them, looked away.

When he reached the stairs, Luis realized he could not ascend with the stroller. As he scanned the platform for an elevator, a man in a suit stopped beside them. He was bald, and muscular beneath his jacket. “I can help,” he said, and Luis nodded thanks, and together the men carried the stroller with the crying baby up the long flight.

CB Anderson ( loves rock-hounding, magenta, little black dresses, snow, scotch, and the stories of Mavis Gallant. Her work has appeared in The Iowa Review, North American Review, Flash Fiction Forward, Brevity, The Christian Science Monitor, Huffington Post, Boston Magazine and elsewhere. A collection, River Talk (C&R Press) was named to Kirkus Reviews’ Best Books of 2014. She lives with her family outside Boston.