A New Bridge Links the Strange Math of Infinity to Computer Science

The archetypal version of this story appeared in Quanta Magazine.

All of modern mathematics is built connected the instauration of acceptable theory, the survey of however to signifier abstract collections of objects. But successful general, probe mathematicians don’t request to deliberation astir it erstwhile they’re solving their problems. They tin instrumentality it for granted that sets behave the mode they’d expect, and transportation connected with their work.

Descriptive acceptable theorists are an exception. This tiny assemblage of mathematicians ne'er stopped studying the cardinal quality of sets—particularly the unusual infinite ones that different mathematicians ignore.

Their tract conscionable got a batch little lonely. In 2023, a mathematician named Anton Bernshteyn published a heavy and astonishing transportation betwixt the distant mathematical frontier of descriptive acceptable mentation and modern machine science.

He showed that each problems astir definite kinds of infinite sets tin beryllium rewritten arsenic problems astir however networks of computers communicate. The span connecting the disciplines amazed researchers connected some sides. Set theorists usage the connection of logic, machine scientists the connection of algorithms. Set mentation deals with the infinite, machine subject with the finite. There’s nary crushed wherefore their problems should beryllium related, overmuch little equivalent.

“This is thing truly weird,” said Václav Rozhoň, a machine idiosyncratic astatine Charles University successful Prague. “Like, you are not expected to person this.”

Since Bernshteyn’s result, his peers person been exploring however to determination backmost and distant crossed the span to beryllium caller theorems connected either side, and however to widen that span to caller classes of problems. Some descriptive acceptable theorists are adjacent starting to use insights from the machine subject broadside to reorganize the scenery of their full field, and to rethink the mode they recognize infinity.

“This full clip we’ve been moving connected precise akin problems without straight talking to each other,” said Clinton Conley, a descriptive acceptable theorist astatine Carnegie Mellon University. “It conscionable opens the doors to each these caller collaborations.”

Broken Sets

Bernshteyn was an undergraduate erstwhile helium archetypal heard of descriptive acceptable theory—as an illustration of a tract that had erstwhile mattered, past decayed to nothing. More than a twelvemonth would walk earlier helium recovered retired the prof had been wrong.

In 2014, arsenic a first-year postgraduate pupil astatine the University of Illinois, Bernshteyn took a logic people with Anush Tserunyan, who would aboriginal go 1 of his advisers. She corrected the misconception. “She should instrumentality each the recognition for maine being successful this field,” helium said. “She truly made it look that logic and acceptable mentation is this glue that connects each antithetic parts of math.”

Descriptive acceptable mentation dates backmost to Georg Cantor, who proved successful 1874 that determination are antithetic sizes of infinity. The acceptable of full numbers (0, 1, 2, 3, …), for instance, is the aforesaid size arsenic the acceptable of each fractions, but smaller than the acceptable of each existent numbers.

At the time, mathematicians were profoundly uncomfortable with this menagerie of antithetic infinities. “It’s hard to wrapper your caput around,” said Bernshteyn, who is present astatine the University of California, Los Angeles.

Partly successful effect to that discomfort, mathematicians developed a antithetic conception of size—one that described, say, however overmuch magnitude oregon country oregon measurement a acceptable mightiness occupy, alternatively than the fig of elements it contained. This conception of size is known arsenic a set’s “measure” (in opposition to Cantor’s conception of size, which is simply a set’s “cardinality”). One of the simplest types of measure—the Lebesgue measure—quantifies a set’s length. While the acceptable of existent numbers betwixt zero and 1 and the acceptable of existent numbers betwixt zero and 10 are some infinite and person the aforesaid cardinality, the archetypal has a Lebesgue measurement of 1 and the 2nd a Lebesgue measurement of 10.

To survey much analyzable sets, mathematicians usage different types of measures. The uglier a acceptable is, the less ways determination are to measurement it. Descriptive acceptable theorists inquire questions astir which sets tin beryllium measured according to antithetic definitions of “measure.” They past put them successful a hierarchy based connected the answers to those questions. At the apical are sets that tin beryllium constructed easy and studied utilizing immoderate conception of measurement you want. At the bottommost are “unmeasurable” sets, which are truthful analyzable they can’t beryllium measured astatine all. “The connection radical often usage is ‘pathological,’” Bernshteyn said. “Nonmeasurable sets are truly bad. They’re counterintuitive, and they don’t behave well.”

This hierarchy doesn’t conscionable assistance acceptable theorists representation retired the scenery of their field; it besides gives them insights into what tools they tin usage to tackle much emblematic problems successful different areas of math. Mathematicians successful immoderate fields, specified arsenic dynamical systems, radical theory, and probability theory, request accusation astir the size of the sets they’re using. A set’s presumption successful the hierarchy determines what tools they tin usage to lick their problem.

Descriptive acceptable theorists are frankincense similar librarians, tending to a monolithic bookshelf of antithetic kinds of infinite sets (and the antithetic ways of measuring them). Their occupation is to instrumentality a problem, find however analyzable a acceptable its solution requires, and spot it connected the due shelf, truthful that different mathematicians tin instrumentality note.

Making a Choice

Bernshteyn belongs to a radical of librarians who benignant problems astir infinite sets of nodes connected by edges, called graphs. In particular, helium studies graphs that person infinitely galore abstracted pieces, each containing infinitely galore nodes. Most graph theorists don’t survey these kinds of graphs; they absorption connected finite ones instead. But specified infinite graphs tin correspond and supply accusation astir dynamical systems and different important kinds of sets, making them a large country of involvement for descriptive acceptable theorists.

Here’s an illustration of the benignant of infinite graph that Bernshteyn and his colleagues mightiness study. Start with a circle, which contains infinitely galore points. Pick 1 point: This volition beryllium your archetypal node. Then determination a fixed region astir the circle’s circumference. This gives you a 2nd node. For example, you mightiness determination one-fifth of the mode astir the circle. Connect the 2 nodes with an edge. Move the aforesaid region to a 3rd node, and link it to the erstwhile one. And truthful on.

If you determination one-fifth of the mode astir the ellipse each time, it’ll instrumentality 5 steps to get backmost wherever you started. In general, if you determination immoderate region that tin beryllium written arsenic a fraction, the nodes volition signifier a closed loop. But if the region can’t beryllium written arsenic a fraction, the process volition spell connected forever. You’ll get an infinite fig of connected nodes.

But that’s not all: This infinitely agelong series forms lone the archetypal portion of your graph. Even though it contains infinitely galore nodes, it doesn’t incorporate each the points connected the circle. To make the different pieces of the graph, commencement astatine 1 of those different points. Now determination the aforesaid region astatine each measurement arsenic you did successful the archetypal piece. You’ll extremity up gathering a 2nd infinite series of connected nodes, wholly disconnected from the first.

Do this for each imaginable caller starting constituent connected the circle. You’ll get a graph consisting of infinitely galore abstracted pieces, with each portion made of an infinite fig of nodes.

Mathematicians tin past inquire whether it’s imaginable to colour the nodes successful this graph truthful that they obey definite rules. Using conscionable 2 colors, for instance, tin you colour each node successful the graph truthful that nary 2 connected nodes are the aforesaid color? The solution mightiness look straightforward. Look astatine the archetypal portion of your graph, prime a node, and colour it blue. Then colour the remainder of the piece’s nodes successful an alternating pattern: yellow, blue, yellow, blue. Do the aforesaid for each portion successful your graph: Pick a node, colour it blue, past alternate colors. Ultimately, you’ll usage conscionable 2 colors to execute your task.

But to execute this coloring, you had to trust connected a hidden presumption that acceptable theorists telephone the axiom of choice. It’s 1 of the 9 cardinal gathering blocks from which each mathematical statements are constructed. According to this axiom, if you commencement with a clump of sets, you tin take 1 point from each of those sets to make a caller set—even if you person infinitely galore sets to take from. This axiom is useful, successful that it allows mathematicians to beryllium each sorts of statements of interest. But it besides leads to unusual paradoxes. Descriptive acceptable theorists debar it.

Your graph had infinitely galore pieces. This corresponds to having infinitely galore sets. You chose 1 point from each set—the archetypal constituent you decided to colour bluish successful each of the pieces. All those bluish points formed a caller set. You utilized the axiom of choice.

Which leads to a occupation erstwhile you colour the remainder of the nodes successful alternating patterns of bluish and yellow. You’ve colored each node (which has zero length) separately, without immoderate knowing of however nodes subordinate to 1 different erstwhile they travel from antithetic pieces of the graph. This means that you can’t picture the acceptable of each the graph’s bluish nodes, oregon the acceptable of each its yellowish nodes, successful presumption of magnitude either. In different words, these sets are unmeasurable. Mathematicians can’t accidental thing utile astir them.

To descriptive acceptable theorists, this is unsatisfying. And truthful they privation to fig retired a mode to colour the graph successful a continuous way—a mode that doesn’t usage the axiom of choice, and that gives them measurable sets.

To bash this, retrieve however you built the archetypal portion of your graph: You picked a node connected a ellipse and connected it to a 2nd node immoderate region away. Now colour the archetypal node blue, the 2nd yellow, and the full arc betwixt them blue. Similarly, colour the arc betwixt the 2nd and 3rd nodes yellow. Color the 3rd arc blue. And truthful on.

Soon, you’ll person made it astir wholly astir the circle—meaning that you’ve assigned a colour to each the nodes successful your graph but for the ones that autumn successful a small, leftover segment. Say the past arc you colored was yellow. How bash you colour this final, smaller segment? You can’t usage blue, due to the fact that these nodes volition link to nodes successful the archetypal arc you colored blue. But you besides can’t usage yellow, due to the fact that these nodes link backmost to yellowish ones from the erstwhile arc.

You person to usage a 3rd color—say, green—to implicit your coloring.

Still, the sets of blue, yellow, and greenish nodes you extremity up with are each conscionable pieces of the circle’s circumference, alternatively than the scatterings of points you ended up with erstwhile you utilized the axiom of choice. You tin cipher the lengths of these sets. They’re measurable.

Descriptive acceptable theorists truthful spot the two-color mentation of the occupation connected the lowest support successful their hierarchy (for unmeasurable sets), portion the three-color occupation goes connected a overmuch higher support of problems—ones wherever tons of notions of measurement tin beryllium applied.

Bernshteyn spent his years successful postgraduate schoolhouse studying specified coloring problems, shelving them 1 by one. Then, soon aft helium finished his degree, helium stumbled connected a imaginable mode to shelve them each astatine once—and to amusement that these problems person a overmuch deeper and much mathematically applicable operation than anyone had realized.

Round by Round

From clip to time, Bernshteyn enjoys going to machine subject talks, wherever graphs are finite and correspond networks of computers.

In 2019, 1 of those talks changed the people of his career. It was astir “distributed algorithms”—sets of instructions that tally simultaneously connected aggregate computers successful a web to execute a task without a cardinal coordinator.

Say you person a clump of Wi-Fi routers successful a building. Nearby routers tin interfere with each different if they usage the aforesaid connection frequence channel. So each router needs to take a antithetic transmission from the ones utilized by its contiguous neighbors.

Computer scientists tin reframe this arsenic a coloring occupation connected a graph: Represent each router arsenic a node, and link adjacent ones with edges. Using conscionable 2 colors (representing 2 antithetic frequence channels), find a mode to colour each node truthful that nary 2 connected nodes are the aforesaid color.

But there’s a catch: Nodes tin lone pass with their contiguous neighbors, utilizing alleged section algorithms. First, each node runs the aforesaid algorithm and assigns itself a color. It past communicates with its neighbors to larn however different nodes are colored successful a tiny portion astir it. Then it runs the algorithm again to determine whether to support its colour oregon power it. It repeats this measurement until the full web has a due coloring.

Computer scientists privation to cognize however galore steps a fixed algorithm requires. For example, immoderate section algorithm that tin lick the router occupation with lone 2 colors indispensable beryllium incredibly inefficient, but it’s imaginable to find a precise businesslike section algorithm if you’re allowed to usage three.

At the speech Bernshteyn was attending, the talker discussed these thresholds for antithetic kinds of problems. One of the thresholds, helium realized, sounded a batch similar a threshold that existed successful the satellite of descriptive acceptable theory—about the fig of colors required to colour definite infinite graphs successful a measurable way.

To Bernshteyn, it felt similar much than a coincidence. It wasn’t conscionable that machine scientists are similar librarians too, shelving problems based connected however efficiently their algorithms work. It wasn’t conscionable that these problems could besides beryllium written successful presumption of graphs and colorings.

Perhaps, helium thought, the 2 bookshelves had much successful communal than that. Perhaps the transportation betwixt these 2 fields went much, overmuch deeper.

Perhaps each the books, and their shelves, were identical, conscionable written successful antithetic languages—and successful request of a translator.

Opening the Door

Bernshteyn acceptable retired to marque this transportation explicit. He wanted to amusement that each businesslike section algorithm tin beryllium turned into a Lebesgue-measurable mode of coloring an infinite graph (that satisfies immoderate further important properties). That is, 1 of machine science’s astir important shelves is equivalent to 1 of acceptable theory’s astir important shelves (high up successful the hierarchy).

He began with the people of web problems from the machine subject lecture, focusing connected their overarching rule—that immoderate fixed node’s algorithm uses accusation astir conscionable its section neighborhood, whether the graph has a 1000 nodes oregon a billion.

To tally properly, each the algorithm has to bash is statement each node successful a fixed vicinity with a unsocial number, truthful that it tin log accusation astir adjacent nodes and springiness instructions astir them. That’s casual capable to bash successful a finite graph: Just springiness each node successful the graph a antithetic number.

If Bernshteyn could tally the aforesaid algorithm connected an infinite graph, it meant helium could colour the graph successful a measurable way—solving a graph-coloring question connected the acceptable mentation side. But determination was a problem: These infinite graphs are “uncountably” infinite. There’s nary mode to uniquely statement each their nodes.

Bernshteyn’s situation was to find a cleverer mode to statement the graphs.

He knew that he’d person to reuse labels. But that was good truthful agelong arsenic adjacent nodes were labeled differently. Was determination a mode to delegate labels without accidentally reusing 1 successful the aforesaid neighborhood?

Bernshteyn showed that determination is ever a way—no substance however galore labels you determine to use, and nary substance however galore nodes your section vicinity has. This means that you tin ever safely widen the algorithm from the machine subject broadside to the acceptable mentation side. “Any algorithm successful our setup corresponds to a mode of measurably coloring immoderate graph successful the descriptive acceptable mentation setup,” Rozhoň said.

The impervious came arsenic a astonishment to mathematicians. It demonstrated a heavy nexus betwixt computation and definability, and betwixt algorithms and measurable sets. Mathematicians are present exploring however to instrumentality vantage of Bernshteyn’s discovery. In a insubstantial published this year, for instance, Rozhoň and his colleagues figured retired that it’s imaginable to colour peculiar graphs called trees by looking astatine the aforesaid occupation successful the machine subject context. The effect besides illuminated which tools mathematicians mightiness usage to survey the trees’ corresponding dynamical systems. “This is simply a precise absorbing experience, trying to beryllium results successful a tract wherever I don’t recognize adjacent the basal definitions,” Rozhoň said.

Mathematicians person besides been moving to construe problems successful the different direction. In 1 case, they utilized acceptable mentation to beryllium a caller estimation of however hard a definite people of problems is to solve.

Bernshteyn’s span isn’t conscionable astir having a caller toolkit for solving idiosyncratic problems. It has besides allowed acceptable theorists to summation a clearer presumption of their field. There were tons of problems that they had nary thought however to classify. In galore cases, that’s present changed, due to the fact that acceptable theorists person machine scientists’ much organized bookshelves to usher them.

Bernshteyn hopes this increasing country of probe volition alteration however the moving mathematician views acceptable theorists’ work—that they’ll nary longer spot it arsenic distant and disconnected from the existent mathematical world. “I’m trying to alteration this,” helium said. “I privation radical to get utilized to reasoning astir infinity.”

Original story reprinted with support from Quanta Magazine, an editorially autarkic work of the Simons Foundation whose ngo is to heighten nationalist knowing of subject by covering probe developments and trends successful mathematics and the carnal and beingness sciences.