Types of Infinity
by Audrey Moyce
When I was a child, I was fascinated by the thought of being a totally different me. The me whose mom didn’t make me fertilize the roses with fish emulsion on Saturdays, but also the me who was actually my cousin, or a butterfly, or a potato. I especially liked thinking about being a potato.
As I grew older, I wanted to try on as many different identities as possible. I hated knowing that any choice I made would take me further away from all the other possibilities. Every yes a thousand no’s.
In order not to foreclose any opportunities, I made choices that were either impermanent or not all that committal. Choices I thought I could “undo.”
In 1874, mathematician Georg Cantor determined that infinity came in different sizes. Matt’s research started with this as its most basic premise, although that was about as far as I ever seemed able to follow him at the time.
On our first date, I took Matt to a café in Chelsea that served both coffee and beer, so I could stay sober if it went poorly and get buzzed if it didn’t. He chose beer after a stilted hug hello. I did the same.
Matt described his research to me, on different sizes of infinity. He charmed me by drawing figures on a napkin to illustrate, then mentioned the theorem that had just earned him a postdoc out in California. I felt cool for finding it cool. At the same time, I wasn’t sure I found it that cool.
But I had been thinking about changing habits. Waiting to be pursued left so much in the hands of others. Over a second round, I asked Matt what he liked to read. Mostly comics, he said, and he launched into lists of titles, series, characters. Then he stopped himself. “I don’t want to bore you.”
“Why don’t you let me decide if I find this interesting or not?” I countered. I felt pleased with the fact that I’d said it, more than I was certain that I meant it.
I watched myself guide the date, finally suggesting we leave. I stopped to kiss him before we descended into the subway, as though the outcome would determine whether or not I would bring him home with me. The truth was I had already decided. What was one night with a stranger?
I sit in sweat on the floor, boxes piled around me like a fort. I relish the ones with a single lamp or set of bowls to take out: one more box erased.
White walls, white blinds, a sea of white carpet. Sterile, like no one’s ever lived here. Water pipes run along the east-facing wall, and by midmorning the cold faucet is boiling. Matt left to see the Hollywood sign with his parents an hour ago.
It’s not too late, I think. Nothing’s even unpacked.
He asked me in my bedroom in Harlem, the two of us happily squished into a twin bed.
“I know it hasn’t been that long,” I nodded, he accelerated. “But it’s not like either of us has ever done something like this.”
I matched his energy. “And I wouldn’t be abandoning my life for you. I’ll probably be in grad school next year anyway!”
“I guess I just want to share this next step of my life with you,” Matt finished. Or maybe he said, “with someone.”
The thought of leaving passes, like it came from a different person. The thought someone in a movie would have.
Matt doesn’t want to run the air conditioning that much, so I feed my fingers between vertical slats to open a window. The blinds slap loudly in the breeze.
Infinity is not a number but a concept. Matt explained that this distinction made his work more like philosophy than math. I liked that. Later, Matt also suggested that writing mathematical proofs was not that different from creating art. I liked that, too.
“We really have to clean the bathroom,” Matt says, drying his hands on his shorts. These pragmatic proclamations please me, seem emblematic of him being a logician. I ignore the upkeep of daily life as though I can will it to disappear.
“I know, but we said we’d do it together,” I whine. My job has become greeting members of an upscale gym. The hours are irregular and the pay is low, but Matt has offered to cover the rent. His salary is nearly triple what he made in grad school, and as he pointed out, if I hadn’t come out here with him he would be paying it all anyway. Still, I applied to every job I saw when we arrived, and I pay for groceries and utilities—I’m no freeloader.
Matt suggests we put on Tig Notaro to make cleaning fun. Tomorrow? I ask, and steer him to our loveseat, then climb onto his lap with my shins on each side of his thighs to face him. I know this pose is childish, infantilizing, even, but when I hook my arms under his and squeeze, the snugness feels too good for me to care.
“Frozen pizza tonight?” I murmur into his wavy hair.
“Can you even call it pizza if it has no cheese?” he jokes.
“You’ll be fine, you can drink that porter.”
“It’s a stout, not a porter,” he says, “but okay.” I roll my eyes in jest.
You are real, I think while studying his face. This is really real.
“You’re really… you,” I say instead.
“Okay, weirdo,” he smiles.
Our first date was my doing, but Matt initiated the second and third. We knew that continuing to date didn’t make much sense, but it also didn’t seem like we had much choice in the matter. We were swept up in something, intoxicated.
At the same time, I ignored the moments that didn’t fit this feeling of momentum. Pauses when we sought the next thing to talk about. Misfires between what one meant and the other heard. I chose to believe that I had been tasked with something, almost nobly so. Some people are just awkward.
Think about numbers as belonging to sets. Imagine that I start with the set of natural, or counting numbers: one, two, three, et cetera. This set is infinite: if I counted for the rest of my life, I would die before I finished, but—crucially—if I could live forever, I would always know which number came next. This infinity is predictable, that is to say, countable.
The Southern California heat in our new town seems unnatural, constant and apocalyptic. I spiral into afternoon panics about global warming and how little I’ve done with my life, but the mornings I like. I wake early, easing the bedroom door shut behind me, do the dishes from the dinner Matt cooked last night, and brew coffee in my French press. Tiptoe outside to our patio and sit on the ground, leaning against the black metal railing as the rising sun warms my back. My mug sits between my bare feet, the gentle scrape of concrete beneath them. I read my book, pausing now and then to relish the newfound joy that here I can tolerate, even enjoy, washing a dish.
Now imagine I don’t start with the first natural number. Suppose that, instead of starting at one, I choose one-half. Where should I go next? Two-thirds? Three-quarters?
Or maybe I choose one and one-tenth (1.1). Perhaps I should go to one and two-tenths (1.2), or would it be one and eleven-hundredths (1.11)?
Or suppose I don’t want to exclude any number. Not even all those unusual numbers, like π, and e, and the golden ratio, and √2, and for that matter any other real number that goes something-point-something-something-whatever. How should I order this bunch of numbers into any sort of list? How do I count?
The gym job redeems itself through the expensive shower products, eucalyptus towels, and free yoga. Their fitness schedule is packed with acrobatic teachers, and nothing feels better than class—wringing myself out, sweating from every pore, sustaining poses until I shake.
The other front desk girls are all actors. I want to write about theater, so we have a lot to talk about but can’t decide if we like each other. When I tell one of them I haven’t shaved my legs in over a year, she says, “Oh, Audrey,” like a tragedy has befallen me and I don’t even know it.
So, the set of all real numbers is not just infinite, it’s uncountable. A countable set, despite being infinite, is smaller than an uncountable one.
Matt and I left New York at the end of July, my lease expired and our relationship six months old. He drove us up through Washington Heights, across the bridge, and onto the New Jersey Turnpike. Only when we left the state did I realize the magnitude of my decision, of who and what I was leaving behind. Thick greenery flew by as my eyes filled with tears. It occurred to me that despite our many plans, a part of me never thought Matt and I would go through with this. This is what you want, I urged myself. Unusual, maybe, but that’s you.
My eyes dried as we racked up the miles. I yearned to turn back, but I also vibrated with a need to keep going. Like with each successive mile I could shed a bit more pain I’d acquired in New York. Like it was tied to the place instead of to me.
I have no car in Pasadena, but I come to love walking. Despite the intense sunshine, despite the stares and raised eyebrows. As I walk to work, I speak into my phone, drafting statements of purpose for my grad school applications. I picture myself back in New York next year. Triumphant this time.
At the same time, when I think about another cross-country move, I’m suddenly exhausted. I’ve only just caught my breath.
It is not just that the set of real numbers is larger than the set of natural ones. There are other uncountable sets even larger than the reals. Georg Cantor ultimately proved that there are infinitely many infinities. Multitudes of multitudes.
Learning this feels like learning, as a child, that every star in the enormous universe has its own universe of atoms inside. Like the reveal in Men In Black, where a galaxy lives in a marble on the collar of a cat.
As I teen, I loved thinking about different sorts of unendingness. The immortality of Greek gods or (as I believed then) my loved ones in heaven. All the space in the universe. The happiness Charlie feels in The Perks of Being a Wallflower at finally having friends.
I even took Charlie’s idea for a poem I wrote in my high school’s literary journal. I suggested that, when you find yourself alone on a beach and dwell in its sensations—most notably the sight of the ocean itself—you’re infinite. “Infinity” was also the poem’s title.
The poem was trying to express what I always felt when staring out at the ocean: that in its enormous expanse I didn’t have a self. I didn’t need one, and that was my definition of bliss. No self, no choices, no no’s.
Even on my days off, I’m walking. I act like new cafes are destinations in a foreign country. I get a hint of a tan for the first time in years.
I often visit the library, walking the warm, sleepy mile around Old Town up to the highway. The building is white, mission-style, with a fountain in the courtyard. I buy a scone or muffin at the coffee cart named “Espresso Yourself” and read in the tepid shade.
The vaulted ceilings inside feel like church. The green table lamps remind me of the one in my grandma’s breakfast nook. When I enter the building’s cool interior, I am saturated with ease. I scour the library’s small academic databases to find criticism for my writing samples, or jot things down until my mind empties.
When something reminds me of New York, it feels so distant from me now. It is almost like there is this pale, other Audrey still living there, a sad hermit in her single room. Here, I am still lonely—Matt works in his office most days, and I haven’t really tried to make friends—but this is another flavor of solitude. Expansive. Self-imposed. Sexy. I wander Pasadena’s suburban streets like it is a deserted wasteland. I chase self-discovery like I am the first person to do it.
Countability helps explain the existence of multiple infinities, but this is different from its proof. Georg Cantor’s proof uses what he called his Diagonal Argument. I can understand it now for flashes at a time, but not long enough to teach it to someone else.
Application deadlines loom as January approaches. Gym members complain about the cold, which hovers around 60 degrees, and I finish my research statements and writing samples.
One lazy day between Christmas and New Year’s, Matt and I stumble upon a discussion about the unfeasibility of the afterlife. He points out the many ways in which its very premise is absurd. What age would we be up there? The age when we died? When we were happiest, perhaps? What if all you want is to see your dad again, and you arrive only to find he’s an eternal eight years old?
Matt’s logic is sound, but a part of me so badly yearns for there to be more than this. The levity with which Matt speaks about it all suggests that he barely knows me.
I get a second job, as a hostess at a wine bistro run by an alcoholic. My self-worth suffers a slow but steep descent as patron after patron looks past me, seeing nobody. This is temporary, I tell myself, not my real life.
But then my heart pounds. What if every school rejects me? Then is this my life?
Before I left New York, a friend flat-out told me I shouldn’t go.
“I hope you aren’t just doing this because you like the idea of being the sort of person who does something like this,” he finally said.
It stung, how wrong he was. It barely felt like a choice, me doing this. My own need—for love? for escape? did it matter?—was forcing me down this path. I would have laughed if I weren’t so upset. I wished I wasn’t the sort of person who did things like this.
All month I wait to hear from schools. I grow ornery. I come home from work feral and pick fights with Matt. I shout one thing and then its opposite, then get angry when he doesn’t reply. I tell him it feels like we aren’t even dating, we’re just roommates who happen to be sleeping together.
“Why won’t you yell at me?” I demand. “I want you to fight me back!”
But he won’t. He says he doesn’t even know what I mean when I say that.
I’m angry that I can’t go down to my old bodega and buy cigarettes. I’m angry that I don’t smoke anymore because Matt hates it. I’m angry that I’m angry and have no solution, or that Matt has no solution for me. Isn’t that what being in this relationship is for?
And if not, what is it for?
Matt showed me diagonalization on that first date. I thought I understood him at the time, but when I read about it for this essay I discover I actually hadn’t at all. I squint at my computer and let YouTube explanations play and replay. I keep waiting for the feeling of I’ve learned this before, but it never comes.
My first rejection comes from New York. Of course, I think. New York doesn’t care about me. When Matt and I met, I was working minimum wage as a theater tech six days a week. I had thought being close to art would inspire me, but all I could see was my own failure to launch. Those who can’t do, hang lights. Matt opened the door at a party in Brooklyn I nearly bailed on in January, the very day he had been offered his postdoc. He was ecstatic and hopeful about the academic life, and it felt like fate. Just a month earlier, I had collapsed on the floor of my mom’s house, unable to stop crying until I had the epiphany that a funded doctoral program was my best, maybe only, shot at happiness.
Emails keep coming: waitlisted for one program, rejected from another. Panic calls like a siren at the hour between afternoon and night. I get in Matt’s car and aimlessly drive. The thrum of the engine revving up and down the nearby hills massages my body, but it fails to quiet my mind. I try to envision myself staying here indefinitely.
But I have no one here, I think, face and neck suddenly hot. The thought of my family and friends makes me suddenly embarrassed. What they all must think of me, with my odd, small life.
I start going to yoga daily. At the end of each session, I lie in corpse pose as sunlight warms my closed eyelids. I envision my body is a glass receptacle dripping paint, all the bright pigment slowly leaking out of me and through the crack in the studio door.
What I can say about diagonalization is that you prove one infinity is larger than another by formulating a number from one set that can’t fit inside the other one. In other words, it is much easier to show one thing that doesn’t belong than it is to find all of the things that do.
We might argue that there are infinitely many lives that would make us happy, but all we can definitively know is which ones, having experienced enough of and moved on from them, do not do so.
Matt hears my phone buzz while I’m in the shower.
“Baby, I think you need to see this!” he says with rare enthusiasm. I dry a hand and stick it out so he can hand me my phone. I see Congratulations, delighted, five years of funding. I step out into the living room in my towel, dampening the carpet.
“I got in off the waitlist!” I shout. Dread falls off my shoulders like a coat. It’s not New York, but it’s a great program, and just a day’s drive north. Maybe I’m out of the abyss after all.
The summer between New York and California, Matt and I stopped over in Iowa. We sat around his parents’ house all August, and I hated how the thick heat kept me lethargic and loathe to exercise. The roll of stomach protruding over my most forgiving pair of jeans said I’d let myself go, that I was sliding rapidly into a pit from which I would never emerge.
One afternoon, I couldn’t stand it. I threw on running shorts and went out in the heat, instantly sweating. I ran, then jogged, then dragged my feet to a nearby playground. My entire body throbbed, blood thudding in my ear.
I felt a yank from nowhere, right behind my belly button. What are you doing here? the yank said. Forget Matt, forget everything. Just. Leave.
But I can’t came the mechanical response. What was I supposed to do, call my mom to buy me a plane ticket? Charge a bus seat to a credit card I couldn’t pay off? Give unrelenting New York just once more chance?
I could have done any of those things. I wasn’t really stuck. But turning back now would out me as a person who, in seeking and desiring everything, ultimately chose nothing. I was allowed to make a huge, impulsive decision without being an idiot, but only if it didn’t announce itself a failure immediately afterward.
I have read that Georg Cantor spent the last years of his life in mental hospitals. It seems at least partially true that he suffered from a lack of professional recognition, from years of disbelief and harsh criticism from his colleagues.
This seems plausible, but so does a nervous breakdown as the result of staring at a schema of the world in which everything is more complicated than you thought, and you can’t even fathom how much more complicated.
Even worse, the ideas that led you to this conclusion all come from you.
Once I’m in school a few months, I break up with Matt. He says over Skype that he feels he isn’t “bohemian” enough for me. I watch him wipe away a tear and remember a story he told me once, about how he had to have surgery on his eyelid when he was a baby because it was too droopy. I suddenly want to take it all back, but I don’t. Once we hang up, I am glad.
My mom and I drive down south to pick up the things I’ve left behind over the summer. I’m angry at having to do this, at having to muck around in the aftermath of my failure, which for some reason I thought would be quick and painless to mop up.
Before I leave, Matt asks through tears if we can talk. I say no, needing to get away from all this. I don’t understand his inability to move on. We never acted like this was meant to last forever—we practically acknowledged its likelihood of failure. His tears after all these months suggest a gap of understanding so huge between us that I fear I have somehow tricked him this whole time. That maybe I wanted it to go this way from the start.
Matt’s research went far beyond Cantor, way past anything I can understand. His subfield was called set theory, as in, take the two sets in this essay—the naturals and the reals—and add more, weirder ones I can’t describe. Make them more and more theoretical, and then see how they behave. Zoology of the fantastical numbers.
The ontological status of these sets is hotly debated. At a departmental dinner Matt and I once attended, his advisor admitted to his colleagues that he was a mathematical platonist: he thought ideas, specifically numbers, existed outside our minds. The other professors were formalists—they believed the opposite. Or maybe the roles were reversed. The point is, it was a big scandal to all the mathematicians, and I couldn’t see what the big deal was.
I still like hypotheticals. What if I were a psychiatrist, a welder, a consultant? I can see the outfits, the accessories, the colleagues and office squabbles. I want to know them all, make them visceral with the experience of it.
And I still see the appeal of self-abnegation. But that poem I wrote years ago (When waves far and near / Extend their tumblings in invitation… [sic]) was only half the story. The ocean I always stared at was on a beach at the end of my grandma’s block. It was usually during holidays, and staring out at sea was a nice reprieve from all the hubbub. I could return to my large, loving family at any moment.
As it turned out, a pocket-sized infinity was what I liked, not the idea taken to its absolute conclusion. I couldn’t be an uncountable set.
Audrey Moyce’s work has appeared in Hobart, Culturebot, The Coachella Review, and elsewhere. She has an essay on queerness in a Game of Thrones collection (Vying for the Iron Throne, McFarland Books).