Infinity Could Not Be Smaller

In Portrait of the Artist as a Young Man James Joyce punctuates his image of the eternities with exclamation marks:

“For ever! For all eternity!... Try to imagine the awful meaning of this. You have often seen the sand on the seashore. How fine are its tiny grains! And how many of those tiny little grains go to make up the small handful which a child grapes in its play. Now imagine a mountain of that sand, a million miles high, reaching from the earth to the farthest heavens, and a million miles broad, extending to remotest space, and a million miles in thickness.... And imagine that at the end of every million years a little bird came to that mountain and carried away in its beak a tiny grain of that sand. How many millions upon millions of centuries would pass before the bird had carried away even a square foot of that mountain, how many eons upon eons of ages before it had carried away all? Yet at the end of that immense stretch of time not even one instant of eternity could be said to have ended. At the end of all those billions and trillions of years eternity would have scarcely begun.”

Vertiginous, I sympathize with Joyce. Still I wonder, what have the most recent few millennia of human ponderings about the incomprehensible vastness of the eternities produced except a bit more than swooning bewilderment and dizzying headaches?

It is that “bit” that interests me today. (A bit is a small amount, a pointless jest, and a tact device for steering a powerful force like a horse or the infinite.) Namely, the conceit of this essay holds that, despite our blurry telescoping images of endlessness, infinity could not in fact be smaller. It is not itself small but compared to what we often think of it as, it could not be smaller than it is, for infinity is a set of names for a very limited yet endless order. Infinity is endless by definition and that definition could not be more limited.

There are plenty of ways to misunderstand infinity: this essay tries to short circuit a bunch of them. Infinity is not a lot of things: it is not a number, not a number bigger than other large numbers (here’s a fun Joycean recipe for 52!), not the biggest number, not too much, not more than anything, not everything, and not more than everything. Here’s one more: I am not claiming that since infinities can always be bigger (they can’t), they cannot be smaller. I also do not mean that since infinity has no upper bound (so far so true but it does have bounds), a thing without a fixed upper bound cannot be made smaller. I argue for none of these. No doubt there are still other unhelpful ways to think about infinity, a few probably in this essay by accident.

I am claiming that modern minimal infinities (the central term this essay explores and clarifies) cannot be any other size—no smaller, no bigger—because endlessness itself arrives on our modern desks nicely packaged, bound, and limited. Just think of infinity like this: if endlessness extends in one direction, that means all the other directions have been nicely tied up. The curious gifts of boundedness can do good work if we learn to let them. This essay aims to make sense of the challenging good news of modern minimal infinities, for the minimal order of infinities could hardly be better news for the latter-day wayfarer, an antidote to headaches, not awe.

On the headaches, I ask the reader’s patience, acknowledging our common middle school trauma in math class. Then again not all middle schools: I have learned much from and dedicate this essay to my mathematically incandescent spouse Kourtney Lambert and our oldest son, Aaron Peters, who prompted this essay in his first year at college. Now that most readers are older, we may take joy and confidence in repeating with Chef Gusteau: anyone can math.

Infinity, David Deutsch suggests, is no more than a beginning. It is a word that refers to nothing except for how we talk about it, and in the last few centuries, a few formal bits—names, symbols, limits, sets, and cardinalities—have helped constrain and domesticate infinity. Modern mathematics has made infinity, while still endless and awesome, minimal. With its tools we can safely step aside from the endless aesthetic abyss and monstrous heavens of ill-understood maximal infinities, finding in its place a more beautiful pluralistic presence: welcome the minimal infinity, an infinity that, because it could not be any smaller, is also enough. Behold infinity: a curiously fine fit for our real, material, limited, changing, pluralistic world. 

Toto, I have a Feeling We are Still in Kansas

Infinite names exist but infinite things cannot. Where does that leave us? Squarely in Kansas. The infinities of metaphysical tornados and totalities have not whisked us and our dear Toto away to Oz. (“Toto” meant everything in Latin before it meant a dog to Dorothy.) It’s not that there’s no available evidence of actual infinite things. It’s that there cannot be such evidence in the physical universe. Our material cosmos operates within the lower bound of Planck minimum units and the upper bounds of space (the edge of light), time (the origin of light, or the big bang), and matter (the speed of light). As Aristotle riffs in Physics and Metaphysics, nothing can be actually infinite since if it is both infinite and actual, then it is not a thing.

In order to contest this claim, theologians, string theorists, and cosmologists have performed great acrobatic feats to twist the infinite around the limits of knowledge and the definition of absolute faith. These contortionists miss the plain point. Physics has no beef with infinities that exist nominally and do good work here and now: indeed, much physics relies on minimal infinities under limits. Our universe contests the fiction of an actual physical infinity but it also invites us to welcome the real material work of nominal minimal infinities. No matter how the cosmological-theological tornados may twirl, Toto stands with all four feet firmly planted on Kansas soil. 

This is fine news if only because actual existing infinities—were they to exist in anything more than the vigorous imaginations of Borges, Joyce, Peck, and Sartre—would be the worst news possible. As this iconic meme about donuts and Homer Simpson reminds, anything fixed forever is hell and everything fixed forever would be total hell. Many scholars, particularly medieval scholastics facing a misreading of Greek, Roman, and (less so) Arabic Enlightenment philosophy, sat deeply with the bottomless terror of the actual infinite. They fought back against the absolute awfulness of a fixed everywhere always forever by inverting the obvious evils of totality and the void: for some, God, being the greatest good, became the only possible actual infinity, a perfect bulwark that crosses Greek greatness with Hebrew goodness against the possibility of anything less. As Thomas Aquinas claimed in Summa Theologica in (roughly) 1273, “nothing which implies contradiction falls under the omnipotence of God.” Thus God became an absolute infinity, omnipresent, omniscient, and omnipotent existing mysteriously outside of (the obvious contradictions in) space, time, and power—and thus the only total totality. 

I am not alone in thinking this a towering mistake. The scriptorian may know that curious verse “endless punishment is God’s punishment.” It does not read “God’s punishment is endless punishment.” This suggests endless is another name for God, not that punishment is endless. It is not that surprising that the modern imagination for a being that punitively takes up all time and space also mistakenly gendered God a single man. Nor may it be a stretch too far to read that verse as suggesting that, by forcing an absolute endlessness upon those we love (including God), we punish them all (including God). Mourn with me what we have done with endlessness: endless punishment is God’s punishment. How many have been punished by misunderstanding endlessness as (would) a totalizing, absolute single man?   

Neither faith nor reason nor evidence compels me to believe in a god that is so absolutely infinite that, like a monstrous kraken or an absent clockmaker, it fills the horizons so completely I must either believe or disbelieve that god is always everywhere all at once. Easy pass. I see no cause to adopt medieval misreadings or their secular reactions as my own.

Thank God absolute actual infinities do not exist and minimal infinities do. With a bit of formalism (discipline through the strait gates), we rid ourselves of the curse of mistaking medieval omnis for an infinity that cannot be. Not unlike ancient times, the names of infinity and God may now easily drift apart into different classes of meaning: no longer one and the same, nor opposites, they pose neither a unity nor a contradiction. Rather like, say, apples and sound waves, the names of God and infinity are free to interact where and how they will. Readers of this magazine may be tempted to flirt with finitist theodicies while analogizing God and the infinite as this essay proceeds; please feel free to do so while remembering that for two things to be in analogy, they must first be different.

Minimal Infinities Mean Less Than We Think

About two hundred years ago, as nineteenth-century Europe and its colonies welcomed the end of the age of political absolutism and, with it, the decline of absolutist philosophy, a marvelous work and a wonder began to roll forth. By the twentieth century, a few folks still considered themselves neo-Hegelians but no one except the obvious baddies were trying to see the world in absolute terms. Somewhere along the way, our sense for the endless, boundless, and inconceivably irrational became more manageable too.

We can find plenty of everyday evidence, reason, and faithful cause for the work of minimal infinities in our lives. My shampoo-conditioner regime has more discrete steps than the three steps we need to make what Aristotle calls a “potential infinity”: “(1) start with a list of items, (2) add an item to that list, (3) repeat the previous step endlessly.” Those quotes contain a potential infinity.

With a sideways squint or two, the traces of minimal infinities pop up into view like those of a scampering squirrel, not a kraken or clockmaker. It requires attention, alertness, and even a certain lightness of being to spot minimal infinities as they jump between the branches of our attention, nibbling on this or that surprisingly contained thought experiment across the philosophy of religion, epistemology, ethics, social and political theory, theology, and applied science and technology like the GPS and our cell phones. Go ahead and sample curiosities such as Zeno’s paradoxes, Hilbert’s hotel (below), Thompson’s lamp, the Littlewood-Ross paradox, and many other “supertask” problems involving infinite steps in finite time. Most of these thought experiments are remarkably contained. (Here a cosmologist takes a mere eight pages to posit infinite dimensions in spacetime. Minimal indeed!) The minimal infinite is more Puck than Typhon.

Even though we rarely realize it, the casual use of the English word infinite is already a way of saying less, not more. If a friend complains about the “infinite number” of deadends their dating apps offer them, they are saying, in effect, they are done, it is too much, it’s over. If a preacher claims, say, the grace of God is infinite, the effect is to inject a salubrious sense of wonder and awe, but the preacher is not exactly laying out further analysis. They are saying, this is all we need to know; because it’s this much, this is all that can be said. If a child says “I love you infinity” or a tween enthuses “I’m going to make infinity dollars when I grow up,” they too are saying the most that can be said. The typical gesture? Widening our hands as brackets. 

This is clever, even if the casual use is backwards. Our hand brackets are misplaced. Consider how often we use the word to mean “stop, it is too much” or at least “stop, it is more than enough,” when infinity means just the opposite: it literally means “keep going, never stop” or just “endless.” But let’s give casual talk some credit. It makes sense we use the word backwards because talk cannot continue forever. Informal talk about the infinite has to attempt to stop talking about the infinite because it does not yet have the advantage of formal talk, which, by contrast, limits infinity without stopping thinking altogether. We need a language to talk beyond the superlatives but that language is not informal chat or hand bracket gestures. This essay models how we use informal talk to stop talking about infinity but formal talk to limit infinity—and keep the conversation going. 

Google Ngrams tend to show word use increasing over time. But not so with the “infinite.” English speakers still reference the infinite but less and less since the mid 1700s, with a low point during the thirty years of World Wars.

Actual infinity is by definition more than too much; so let’s learn how to talk about it in a few formal ways. Learning to say less so that we can say anything at all is, on balance, a solid step forward. 

Five Ways Infinity Could Not be Smaller: Names, Symbols, Limits, Sets, Cardinalities

Names first. Infinity is a name and names really matter. Try going a day without a name, an ID, login, or a physical address. A distinction: if there exists a set of physical things (physics) and a set of possible names (mathematics), the two intersect in the set of named things. In other words, while physics describes material reality with names (and thus nothing physically nameable is infinite), mathematics does formal work with any name (and thus infinity can be so prevalent it is an afterthought). Thankfully, it’s not an either-or choice. Even though infinities name no things in the realm of physics, and minimal infinities belong squarely to the realm of mathematics, modern people (especially physicists) have learned to use a few bits—names, symbols, limits, sets, cardinalities—to make minimal infinities do good work.

All numbers are names while infinity is a name but not a number. It cannot interact with numbers as interoperable equals. Instead, the infinite is a name for a process or group which is endless, non-terminating, without finish, or in-finite. All names summon things to mind, index meaning, and manage complexity; as a result, names tend to be simpler than the things they describe. For example, calling out “Toto,” Dorothy’s dog’s name, might just summon into her presence a hairy, smelly, and utterly lovely creature bursting with irreducible reality. Just two syllables summon forth a real being. What if we tried to call all real things with a name? For example, what if we tried to summon forth the longest list of items that refers to actual physical spacetime? Perhaps we could imagine, say, “a list of all possible subsets between every point down to the Planck unit in the observable universe.” (Every point everywhere connected in every combination.) This behemothic yet perfectly finite list would require more resources than exist in the universe to write out and yet it takes a mere 18 words to name, or just the bit in the quotes above (check out the Berry paradox for fun). Behold the awesome power of names to invoke and limit. Names are power narrowed into strait gates. So are symbols and sets.

A Symbol: A Simple Stand-In

Consider this symbol, this single-character name. The infinity sign. ASCII 236. An eight on its side. A two-dimensional Mobius strip. The lemniscate may come from the last Greek letter omega ω by pulling together the open rings in the last letter in the Greek alphabet. (The lower-case letter, not the upper-case letter, here does not weaken the less-is-more vibe of this essay.) In other words, the symbol for infinity performs a potential minimal infinity by taking the smallest last letter in the alphabet and then looping it. The lemniscate resembles a potential infinity in the making. The end result? Like the idea, the symbol for endlessness is super short. With the loop of a pencil, the endless has been named and contained. Modern calculus notation, in fact, squirrels away the infinity sign (lemniscate) into small subscripts and superscripts. A small symbol for an endless thing. Behold infinity, an afterthought. 

A Limit: Infinite Division Made Sane

Infinities have limits thanks to calculus: calculus is the art that domesticates not infinity (we’ll talk about set theory in a second) but the reciprocal of infinity, or the infinitesimal (or one divided by infinity). As Amir Alexander documents in this phenomenal history, the infinitesimal proved a dangerous idea. No less than usually peace-preaching monks in medieval Europe murdered each other over the smallest differences between the smallest of the small. For some, the infinitesimal represented mankind’s contribution to God’s plan and so they fought over to understand humanity’s part, the infinitesimal: is our infinitesimal role genuinely nothing or was it just small enough that it must be smaller than all other things but also still a thing? This was early modern Jesuit code for, Are we saved by infinite grace or by infinitesimally small works? Conversely, is the infinite a thing that is so large, it must be a genuinely incomprehensible totality (an actual infinity) or just larger than all other things but still a thing?

Modern life and math answer plainly: the latter. Even though these things cannot exist physically, the infinitesimal and the infinite must both exist nominally as limited limiters, or as actionable names that do work. They cannot be reduced to nothing or everything, respectively; rather they can only exist within limits. A mathematical limit is a finite value that a function can approach infinitely but not reach. Reread that sentence again if you want—it has a cool twist: within finite limits, potential infinities create finite values (or gorgeous shapes like the Koch snowflake in a moment). The introduction of the mathematical limit solved an age-old puzzle: as Steven Strogatz puts it, in order to solve anything continuous, we have to chop it up first into the tiniest bits and then put them back together. So, when adding up a sum of infinitesimals under the curve in calculus, for centuries we had to assume that the smallest bit, an infinitesimal, is both practically zero (so that you cannot divide it any smaller) and at the same time that it is somehow also more than zero (so that when you add all the bits together under a limit, they sum to a finite value). This apparent contradiction was resolved by Bolzano, Cauchy, and Weierstrass in the 1800s with the invention of an infinite sum that converges to a limit, basically rendering calculus into arithmetic. The limit vanquishes Zeno, who used to divide up space to keep Achilles from catching the tortoise or even from moving. With Zeno contained, motion under the curve was made calculable.

The limit does abundant if often hidden work: Strogatz shows how limits, infinitesimals, and calculus powers “cell phones, TV, GPS, and ultrasound. We wouldn’t have unraveled DNA or discovered Neptune or figured out how to put 5,000 songs in your pocket.” With math for curves in motion it makes sense why Mars is sometimes in retrograde. Limits do real work by making the smallest thing—an infinitesimal—manageably a bit less small. (It is not quite right to say bigger since an infinitesimal still needs to be the smallest possible thing.) Thanks to limits, infinitesimals become not too small, infinities become not too big. As in math, so in reality (at least in this sense): Without the limits of ground and heavens, floors and ceilings, and distant horizons under the curve of the earth, life and meaning have no obvious orientation.

Limits work in space and time too, and therefore infinities fit perfectly well into limited spaces and times. Take the Koch snowflake, invented in 1904. It sounds paradoxical at first: a Koch snowflake is a purely mathematical fractal shape whose perimeter is infinite and area is finite. The snowflake’s endless edges, if stretched out, would stretch forever. Instead they fold up so tightly, the snowflake fits in the palm of your hand (or any other finite size). To make a Koch snowflake, make a potential infinity: (1) start with an equilateral triangle, (2) bend the middle of each triangle edge out into a new smaller equilateral triangle, and (3) repeat the last step, adding a new smaller triangle to each edge ad infinitum (more here). It’s not the infinite edge but the finite area about the Koch snowflake that bedazzles. It’s not surprising we cannot measure an infinite perimeter. It is surprising its area is finite. Here’s a simple way to think about how a Koch snowflake has a finite area. You can easily draw a circle around the snowflake, which it would never cross. Since the circle’s area is finite, the snowflake’s area must be finite too.

We are staring at an image whose edge or perimeter tends to infinity. As Mandelbrot and others showed, the length of a coastline depends on the length of one’s measuring stick, or in the case of the image below, eyesight and pixel size. A potentially infinite object is staring right back at us. It’s not that it seems normal; it is uncanny that small infinities are normal (what’s normal can sometimes be awfully strange). Minimal infinities scamper among us.

Sets: Infinity’s Container

Sets are built for managing infinities. Infinite sets pose no problem because sets are no problem. Sets did not emerge into their own theory—especially what we now call standard Zermelo-Fraenkel set theory—until the last hundred and fifty years or so, although the basic idea of a set is as simple as a name for other things in order. A set is the simplest thing that is not just a thing itself; it is a collection or class of other things. It conceptually contains things, gives a name to their group, can give order to them, and (like names and symbols) are simpler than the things they contain. Again, naming the set “all the large mammals on earth” is way easier than doing anything with any one of its members: lions, whales, dolphins, etc. Sets also let us do really basic operations between them: we can find their union, intersection, difference, complements, products, and partitions. Set theory, or the math for managing the infinite, has fewer operations than most calculators.

Sets are so simple that if sets were any less simple, they arguably would not be able to contain infinity; and since infinity is a name that references those formal bits (like sets) that let it do work, that means infinity too could not be any simpler. A set can contain an infinity of things because of how narrow and nameable a set is as a container. That is the magic that lets it contain infinity: a set is a single thing. As we’ll discover in a moment, it can even name types of infinities incomparably larger than other infinities. Naming is real in that it brings new kinds of reference into reality. (My name, as family historians know best, exists in its paperwork and witnessed utterances. As a twist on the old tree in a forest gag, if a name is uttered without a witness does it refer to a thing?) Since naming a set brings a reference into existence, the fascinating history told by Loren Graham and Michel Kantor should not bewilder: still holding to the medieval analogy of God and the absolute infinite, leading mathematicians in the Lusitania seminar for set theory at Moscow State University in the 1920s also believed that, by invoking the name of God in ritual prayer, they were bringing that name into reality. We don’t have to believe that to still see the reasoning behind sets: infinities are names, and sets name infinities into existence. 

The Hilbert Hotel is a famous example of an infinite set. Consider what can and cannot be said about this story, first proposed by the German mathematician David Hilbert almost a century ago in 1924: 

Imagine that you are a hotel manager at a hotel that has an infinite, or unending, number of guest rooms numbered from 1, 2, 3, ad infinitum. When you start your shift today, each room is already occupied with a guest (accept that as an axiom, but if you wonder how, just work by potential infinity under a limit: perhaps your colleagues previously teleported each guest to their new room using an infinite teleporter? You are already in an infinite hotel, so why not an infinite teleporter?). Then, to your consternation, a bus pulls up during your shift with an infinite number of new guests, each wanting their own room in your hotel. What do you do? How, if at all, can you accommodate them? Take a moment to think about it before reading on. 

If you want a hint, remember that the “number” of rooms you have at your disposal is endless and you can ask guests to move from room to room in ordered ways. The image above tells us what to do if one new guest arrives, but what to do with an infinity of new hotel seekers? Read on when you’re ready for a solution.

Solution: it doesn’t make sense to ask all the current guests to move down an infinity of rooms as a single block. One cannot add to infinity because it is not a number. But you, our intrepid hotel manager, can play with the internal order of the two infinite sets: namely, you can ask each current guest already in a room to move to the new room number that is twice their current room number. So the guest in room 1 goes to room 2, the person in room 2 goes to room 4, the one in room 3 goes to room 6, the one in room 4 goes to room 8, etc. (That’s some “etc.”) Now you have an infinite number of occupied even rooms and an infinite number of vacant odd rooms. At last you can now assign each of the infinite new guests from the bus to one of the odd rooms. Double the first guests’ room numbers, and voila, everyone has a room!

Without the set here (the list of guests), infinity would prove inhospitable. With it, it could not be more hospitable, welcoming endlessly.

Cardinality: Infinity’s Order

What’s the point of the Hilbert hotel thought experiment? It is this: we can (almost only) think about the set of infinite hotel rooms through our fifth and final formal bit—the cardinality or number of rooms. In other words, the main thing one can ask about an infinity of rooms is, in what order are the rooms and how can we rearrange things in that order? Cardinality, or the number of elements in a set, asks for the most basic comparison: is set A larger or smaller than set B? That’s it. To paraphrase philosopher of mathematics Markus Pantsar, transfinite infinities rest on an intentional minimization of analysis to the simplest type of size comparison. Even anumeric babies and nonhuman animals can tell that one pile has more items than another (even if those items and thus the pile are smaller). That is what set theory does: it compares set sizes. For example, an endless row of numbered objects is an infinite set, and that’s it: 1, 2, 3, 4.... Its cardinality, and how that cardinality maps onto other sets, is all we can say about it. In this light, infinity is not big or small; it’s a name of such a limited order that everything but a few things about it is senseless.

Often the things that overwhelm us do so because we do not yet understand that they are closer to us than we can see, not outside what we can see. Infinity among others.

The Hilbert Hotel is silent on all the real-world questions: how did infinite folks get there in the first place? How does one manage the guest keys in such a situation? What about all the complaints from the infinite hotel guests being told to move rooms? Can’t the guests just double up? What gas mileage does a bus with infinite seats get? Would the tips to the cleaning staff alone support a new economy?

All of this reality-adjacent thinking must remain mute. Infinity has nothing to say. On the other hand, the moment we move beyond minimal formalisms, culture comes rushing back in: for example, we can ask slightly pesky questions like, Why do most images of Hilbert hotels feature identical man stick figures? Why all the top hats? This smallness, the cultural limits of our representations of formal infinities, is a limit in every sense. Let’s come back to all the dudes later. For now consider: why does so much art about infinity court a modernist aesthetic that appears to wish it were Victorian?

Infinite hotels accommodate infinite guests but make no room for differences among them. All our proverbial hotel manager can say about infinite rooms is something minimal about its internal order: “please move to double your current room number.” 

The Continuum: Infinity’s Endings

In the late 1800s, Georg Cantor discovered transfinite sets, or that there are different sizes, or orders, of infinity. Cantor, a Russian-German set theorist whose name means “Singer,” coined the term “transfinite” out of his lifelong reverence and search for the idea of an actual infinite, the most infinite of them all. His own life, limited by his struggle with bipolar, offers a stirring reminder of our mortal limits.  (For the next bit, let's visualize the number line firmly in our mind. Experts suggest zooming in and out with eyes closed.)

Cantor showed how some infinite sets are the same size and other infinite sets are different sizes. For example, he proved that the size of the set of counting numbers (1, 2, 3...) is the same size as the set of all rational numbers (or all those that can be expressed as M/N where M and N are counting integers and N is not zero). This might surprise at first glance since, presumably, there are a lot more fractions (or rationals: 1/2, 3/5, 4/17, 101/1001, etc. just to mention four fractions between 0 and 1) than there are counting integers (1, 2, 3...). Yet Cantor showed that if we had two infinite sets of all the rational (again, M/N fractions) and counting numbers, we can completely map the one set onto the other set, one pair at a time with no overlap or gaps. In other words, he showed that the two infinities of counting and rational numbers are the same countable size: for example, if our Hilbert hotel manager had two lists in each hand, a room list by counting numbers and a guest list by every rational fraction, the two lists correspond one-to-one. Behold a homeomorphic mapping: each (fraction) guest gets a (whole number) room. 

That’s not all. Here Cantor stayed with the trouble, discovering that other infinite sets are uncountable, or do not correspond one to one with the counting numbers (1, 2, 3...). Let’s make a list of irrationals: for example, the square root of 2, pi, e, etc. (Irrational numbers are those that cannot be expressed as a ratio or that have infinite nonrepeating decimal places; by the way, irrationals are conceptually just as precise as rational numbers: the root of 2, for example, has one precise spot on the numberline, just like, say, 4.82, even though we can’t express root 2 with discrete digits.) For ease, let's consider only the irrational numbers between three and four. This line segment has exactly two counting integers, 3 and 4. It also has countably infinite rational numbers: can we imagine every m/n whose value is on the line segment running from 3 to 4? No, but we can show that its list would be both infinite and countable since, say, (3n+1)/n, where n is the (positive, whole) number of items in the list, gives the list 4/1, 7/2, 10/3, 13/4, 16/5, ad infinitum (all on the line segment 3 to 4). So, a countable infinity fits between 3 and 4.

But then Cantor discovered that this same line segment also contains a third set, an uncountable number of irrationals, of which only one is pi (3.1415926...). How do we know it contains uncountable irrationals (numbers we cannot express as the ratio m/n)? Let’s take pi, or that endless string beginning 3.1415926…. Now, Cantor asks, how might we find the closest number to pi? Or, if this is easier to think through, what’s the number nearest to, say, 3 that is not 3: how close can you get and why? How would we settle on the next neighboring number? (If we could count all the next neighboring numbers, Cantor reasoned, we’d have a countable infinity.) Perhaps we might string out pi’s digits as long as we can and then toggle, say, the last digit up or down one digit: maybe we could turn the six at the end of the string of decimals above into a five or seven, for example? That’s a tiny difference, but is it the smallest? By formally examining the set of all irrationals, Cantor proved—and I think this is pretty cool—that between even the two closest neighboring numbers is not just another irrational but a continuum of other irrationals. Between every two nameable numbers lies an infinity of other unnameables. It’s Unzahl—German for “myriad,” literally un-number—all the way down.

Imagine our Hilbert hotel hallway is our number line. It now has two sizes of infinities in it. There’s the countable series of infinite door numbers that stretch endlessly down the hallway. (Approach it by walking.) But there’s this second irrational type too: if our hallway were a continuum of nameable door fractions and irrational numbers, then just by looking closely at any point on the wall, we would discover an uncountable infinite number of potential door names. (Approach it by staring closely at the wall.) There would be no empty space between the door numbers. The closer you look between two doors, the more door names and gaps between door names would appear, and then even more doors between those doors. Like the animated edge of our Koch snowflake, the hallway wall scrolls ever inward containing both infinitely nameable and unnameable doors. It is doors upon doors all the way down (or countable doors if you keep walking down the hallway and uncountable doors if you microscope zoom into any section of that hallway). The smallest nameable difference in a continuum contains an infinity of infinities within it. The tiniest sliver of any continuum “outnumbers” all nameable digits on the entire number line. This is what continuous means. Continuums are tricky business.

Mathematicians call this irrational for a reason. Note that we can only see how supersaturated a continuum is by zooming in. To see the infinite at work, we have to look closer. That’s the literal point, one after another (think a continuum), of this essay: the smallest possible infinities do the most work. Zoom in, not out. 

The size of uncountably infinite hotel doors is different from the size of countably infinite doors numbered 1, 2, 3.... A list of irrationally continuous guests does not map onto the list of countably infinite hotel rooms. At this point our visual imagination may shipwreck against the rocks of impossibility but be slow to leap to despair or awe. All of our handy bits of formalism for containing infinity keep working. (Hat tip to sets. Sustainable awe lies in small things.) Transfinite sets like these play really well with symbols, names, sets, and cardinalities. Perhaps transfinite sets are a bit like most people in history: hard to visualize because they are well-behaved. Claiming Sephardic paternal grandparents, Cantor even renewed an old symbol to talk about new transfinities: the Greek alphabet, having ended at the looped-omega lemniscate, spilled over into the Hebrew alphabet, with the cardinality of the counting numbers named Aleph nought, ordinal numbers Aleph one, and two to the Aleph nought for the continuum of irrational numbers.

Here are two more examples of transfinite sets: imagine, with Kolmogorov, that you are throwing darts at a target. Each dart hits only one point, and thus only one nameable coordinate, on the target surface, a two-dimensional finite plane. What kind of coordinates are you most likely to hit? Is it countable coordinates like the perfect bullseye (0,0) and all other discrete coordinate links (-0.1, 0.2), on the one hand, or irrational coordinates like (0.7213158..., -0.92364...) that continue with discontinuous digits forever, on the other? Take a moment to consider before reading on. Like the irrational hotel hallway, there is an aleph nought of countably infinite coordinates on the target. But that’s just the beginning: between any two nameable rational coordinates our dart could hit there also lies a higher order (two to the aleph nought) of uncountably infinite points the dart could strike. The dart doesn’t care, so it flies straight into the omnivorous maw of different sizes of infinities and transfinite sets. Here Kolmogorov hints: since our dart hits only one coordinate point, and since the infinite rational target coordinates have a smaller size or cardinality than do the infinite irrational target coordinates (another near bullseye: 0.28551347..., -0.018603704..., etc.), our dart will hit only irrational coordinates. 

Here’s an easier example: imagine a genie grants me, a pasta lover, my wish for an infinite field piled high with cacio e pepe (homemade spaghetti with cheese melted everywhere, sprinkled with pepper flecks). The spread of cacio cheese represents the uncountably infinite, everywhere all at once, while the pepe pepper flecks represent the countably infinite at specific points. With enough time, we could count the pepper, not the cheese. The two represent different sizes of infinities.  

Notice again how these examples involve zooming in to look at and reframe smaller, not bigger, ways of thinking? To make a name a practical idea, it needs disciplining operations—like symbols, limits, sets, and cardinalities—to make it as small as can be. We do not have a stable idea of a thing until after we have the techniques for engaging and practicing it, an idea that Bernard Siegert calls “Kulturtechniken”: for example, counting and accounting practices precede the ideas of formal mathematics, drawing and art crafts comes before art markets and art criticism, shipping-building predates maritime cartography and colonialism, etc. By the practical arts are ideas brought to pass. Without our bits, transfinities make no sense: once we have a few tools in place, though, even the irrationally inconceivable realms like the continuum (hotels with continuous doors, continuous target points, and cheesy pasta fields) become abnormally normal. The stupor fades, thought remains. 

A Minimal Infinity is Enough

Welcome the minimal infinity: a name for endlessness that emerges from the set of (cultural) techniques and formal language—names, symbols, limits, sets, and cardinalities among them. We first practice with and thus make their ideas manageable. It is the modern person’s blessed if often unseen status quo to have infinities calmly sitting amid our pews bustling with so many other everyday names.

How untameable could an idea really be if it can be sketched in a few pages? Ad infinitum and ad nauseum need no longer be near synonyms. Behold Joyce’s mountain, scaled without the exclamation points.

The endless is not beyond thought; as this essay argues, it’s almost the other way around. By using its small, ordered bits for making it minimal, we can only think endlessness into nominal existence and then let it do real-world work. We bump up against limits in our thinking about the infinite not because our thinking is too limited but because the only way to think about it is thanks to limits. 

May we limit, without stopping, our thinking. 

Again, confusing things need not always exceed our view. A thing that exists because our view limits it can confuse too. Infinity is such a thing. It’s a curious risk of modern life: we don’t see only effects; seeing causes things too. This sounds at first a bit tricky because if seeing can cause things, then how do we know the things we are seeing are real? If things exist because of the ways we view them, how do we know it’s not hallucination (insert large language models and simulation theory banter here)? The answer is clear and chill: we learn to see, welcome, and work with limits. Limits that do good work disband the hallucinations and let readers of this magazine get back to the work of, as Mason Kamana Allred put it in a signal brand new book, seeing things. Two cheers for our ability to see limits. They are enough.

Consider the good that limits do. Take the limits of language. How many vainly dream of one day conversing like angels in limitless ESP? No thanks: let’s lean into the very languages we possess—a language made by limits. Imagine if we learned to welcome and master, not apologize or fret about or try to avoid, the limits that make up our language. Public addresses might feature more focused scopes, stirring sparks, clear assumptions, well crafted stories, limited conclusions, and shorter essays. Despite free speech absolutists freely (and reliably unconvincingly) insisting otherwise, the best language is not limitless talk. Perhaps the speech that flows forever is that which first binds itself to serving others, listens carefully, and limits itself. Angels and their modern-day wireless imitators don’t communicate without limit: they model limited communication exquisitely by listening a lot more than one might wish, speaking subtly in specific contexts, and limiting their speech to specific effects. (The actual terror of Alexa hearing and recording my words would easily be exceeded by the endless potential horror if Alexa, like an unlimited angel, heard and spoke my thoughts, except that the first Alexa is real). Sometimes those effects are salutary, sometimes they are surveillance. Are we so sure that infinite and everyday talk are really all that different? 

What can we do with a name? A few powerful things. We can call Toto into our presence. We can address a message to its receiver. Name actions matter so much because they are constrained. (It is not for naught, as Amit Pinchevski and John Durham Peters develop, that media that broadcast messages without specifically named receivers—like radio and television—have twentieth-century associations with madness, radioheads, and thought broadcasting.) Infinity is not small of course, but to claim it could not be smaller means its mischief has been managed: it’s not too big to think; rather all that it is, is thinkable, and because its name and operations are limited, that’s enough. In fact, it’s enough, together with the infinitesimal, to power modern life. Minimal infinities render small marvels in our lives because they have become afterthoughts.  

The story of how the infinite became a good news afterthought is also a story of women: namely, the pluralization and minimization of infinity has begun to make room for women at the table of dialogue about the infinite over the last handful of centuries. Tullia d’Aragona, the sixteenth-century Italian courtesan and poet, argued powerfully that the Platonic orthodoxy of her time made no room for women in the realm of physical bodies and souls, sense and intellect, and that any legitimate Dialogue on the Infinity of Love, in her title, must account for the nature of real living people. Infinity is learning how to hear and embrace pluralistic, multiple voices. There are still many to hear and much to learn. Most infinity talk, like much in the modern public-private dynamic, echoes men's voices expanding into the public and women's voices contracting into private space. This essay is no exception. It benefited from and is dedicated to, as previously mentioned, my spouse, an incandescent mathematical thinker. While no one but I am responsible for the essay’s length (a deliberate choice to perform a bit of endlessness), it was also edited for clarity and improved throughout by three otherwise unmentioned brilliant editors: Rachel Frandsen Jardine, Rachael Givens Johnson, and Charlotte Wilson. Their voices ring out, in unacknowledged waves, across its better moments, and yet who does the essay name? Its intro starts with Joyce’s Young Man. It sideeyes a monopolizing single man god. It is illustrated with myriad top hats staying at Hilbert’s Hotel. Its background Stanford Encyclopedia of Philosophy article on “infinity” mentions “he” 43 times and “his” 30 times and “she” or “her” zero times. Perhaps we can observe other forms of talk that speak with the voice of women in naming so many men? May we seek not to force a balance but to better recognize, recover, and serve the rich plurality hidden by imbalances.

As d’Aragona asked in 1547, “since the word ‘infinite’ can stand for a plurality of things, which of its various significations do you take in this debate?”

As in the world, so with the minimal infinity: plural, embodied, material, in motion, limited, endless. As Pantsar put it, Zeno has no power here. Test it out: I invite the reader to search “infinite,” “eternal,” or “endless” in their favorite source for faithful material, official or informal, religious or otherwise. Review the results and ask whether the topics at hand—hope, faith, mercy, heaven, debt, relationships, law, speech, or whatever else—would work better if the “infinite” here were understood (A) to maximally cover all of spacetime, (B) to casually mean “incomprehensibly big” to spark awe and halt our thinking, or (C) formally to apply many limits to our concept of the endless? (Check out the finitism of the mathematician-apostle Orson Pratt’s monistic materialism while you’re at it.) My bet falls between B and C, leaning toward C.

Consider how minimal infinites disciplines the improprieties of (unlimited, totalizing) maximalist infinities: a minimal sense of infinite grace encourages discernment and contextual judgment whereas a maximal sense of infinite grace might excuse inaction in the face of individual and systemic injustice. To excuse a repeat abuser or an abusive system out of blind trust in a maximal sense of infinite grace forgiving all things and making all right in the end heaps suffering on the abused, dims the abuser’s hope for change, and indeed stubs the graceful work of the infinite. In comparison to endlessness, “seventy times seven” appears a remarkably finite formulation bracketed with symbolic power. Perhaps grace is infinite only when minimally applied. It applies unevenly, locally, variably, and often urgently in a non-flat universe. It, like the minimal infinite, does its vital work within the limits of here and now through names, symbols, limits, sets, and other formal forms of discipline (compasses, squares, rulers, tables, and other daily and weekly rituals that renew and reset us in a bounded infinite flow, etc). Once so circumscribed, minimally infinite grace flows free wherever it may.

What endless things are also good because they are bound? Perhaps the reader will add to or complicate this shortlist of blessedly bounded perpetuities: any endless path in science, learning, and thinking must, in order to avoid spinning out into conspiracy, welcome the checks of careful evidence and interpretation at every step. Or, any legal right achieves total immunity only inside the limits of a specific person, their civic responsibilities, and clear law. (For example, property rights apply to many but not all things: no one legally owns charitable nonprofits in US law. Or, absolute immunity may apply to an office but not to its holder’s actions outside of that office’s perimeter.) Or, to nod to the donut model, an economy worthy of the eternities provides at or above the floor of its inhabitants’ basic needs for medicine, food, water, clothes, and shelter as well as below the ceiling of what threatens making the environment unsustainable for life. 

Maybe we can even see minimal infinities in popular culture? Perhaps consistent top-10 artists like James Brown, U2, or REM approach a minimally infinite stature because (unlike the many one-hit-wonder tangent curves that burn out and unlike the few megastars like the consistent-bestsellers Beatles, Taylor Swifts, and Drakes) these top-10 (not top-1) artists continuously produce song contenders that, like limits to infinity under the curve, always appear “less than” in the public eye of many critics but never stop. What other artists approach such infinitely modest greatness? What other ways might minimal infinities help us see anew?

Adam Miller suggests that one avoids the damnation of eternal return by getting on with our ordinary living; adding bits of formalism, I have argued, enriches the case for everyday living. Minimal infinities carve endless paths between the guard rails against the avalanches of absolutism on one side and the oceans of nonsense on the other. The search for the summum bonum is as much a distraction as is the detour of cynical relativism. The void and totality often spring the same metaphysical idealist traps. Instead by engaging with minimally infinite goods—once bounded into real-world action by our use of formal bits—we invite the infinite that fits our attempts to live and care better for one another, however unevenly, in limited space and time here and now. May we live timely eternal lives here and now thanks to minimal infinities working in our limited time and space. By working with minimal infinities, we neither relinquish nor capitulate to the infinite: we take part in the infinite. We share in the infinite because we all—the infinite too—are partial, limited, and sometimes better for it.

Infinity is a name, not a number. Its name has over the last few centuries opened up ways to make room for more diverse, pluralistic, limited, material, endless things in motion. Because it can be minimal and bounded, its endless work multiplies among us. Perhaps the best way to limit talking about infinity is through its formal bits: the symbols, names, limits, sets, cardinal orders, the occasional seemingly endless but still obviously bounded essay, and other limited actions by which we think, live, and take part in the infinite. A minimal infinity calls us to see how the present might be limited differently. It lets us change sustainably. It is narrowly defined. It is with us here and now.

Infinity, thank God, could not be any smaller. The infinite is just the beginning. 

Benjamin Peters is a Wayfare Associate Editor. He is also a media scholar, author, and editor interested in Soviet century causes and consequences of the Information Age.

Art by Kazimir Malevich, Piet Mondrian, Bart van der Leck, and Theo van Doesburg.