Imagine discovering that the mind-bending mysteries of infinite sets in math are secretly speaking the same language as the practical algorithms powering our computers—what if this hidden link could revolutionize how we understand both? That's exactly what's happening in the world of mathematics and tech, and it's got experts buzzing with excitement.
January 4, 2026 | 7:00 AM
Experts in descriptive set theory dive into the quirky world of infinite math. Recently, they've demonstrated how their intricate puzzles can be rephrased using the straightforward terms of computer algorithms.
Illustration: Valentin Tkach for Quanta Magazine
The initial version of this article can be found here (https://www.quantamagazine.org/a-new-bridge-links-the-strange-math-of-infinity-to-computer-science-20251121/) on Quanta Magazine (https://www.quantamagazine.org/).
At the heart of all contemporary mathematics lies set theory, which explores how to group and manage abstract collections of items. Typically, working mathematicians don't dwell on it while tackling their challenges—they simply assume sets function predictably and move forward. But there's a dedicated group that never lets it out of sight: descriptive set theorists. This tight-knit crew keeps probing the core essence of sets, especially the wild, endless ones that others tend to sidestep.
Their area of study has suddenly found some unexpected companionship. Back in 2023, a researcher named Anton Bernshteyn (https://bahtoh-math.github.io/) unveiled a profound and unexpected tie (https://link.springer.com/article/10.1007/s00222-023-01188-3) between the isolated realm of descriptive set theory and the bustling field of modern computer science.
Bernshteyn proved that questions involving specific types of endless sets can be transformed into issues about how computer networks exchange information. This surprising pathway between the two worlds caught everyone off guard, from math purists to tech innovators. Set theorists communicate through logical frameworks, while computer scientists rely on algorithmic instructions. One side grapples with the boundless infinite, the other with tangible finitude. Why on earth would their challenges overlap, let alone mirror each other perfectly?
"It's downright bizarre," remarked Václav Rozhoň (https://vaclavrozhon.github.io/), a computer science expert at Charles University in Prague. "You wouldn't expect this kind of overlap at all."
Following Bernshteyn's breakthrough, colleagues from both camps have been eagerly testing ways to traverse this new connection, using it to derive fresh theorems on either end, and even stretching it to cover broader categories of challenges. A few descriptive set theorists are now borrowing ideas from the computer science perspective to reshape their field's terrain and revisit their grasp of infinity itself. For beginners, think of it like this: infinity isn't just a big number; it's a concept with layers, and this bridge helps peel them back in ways that feel more approachable, almost like debugging code to understand endless loops.
Anton Bernshteyn has been tirelessly revealing and delving into vital links between set theory and practical domains like computer science and dynamical systems.
Photograph: Siiri Kivimaki
"We've been tackling remarkably parallel issues all along, but without ever chatting directly," noted Clinton Conley (https://www.math.cmu.edu/~clintonc/), a descriptive set theory specialist at Carnegie Mellon University. "Now, it's unlocking a flood of potential partnerships."
The Fractured Foundations of Sets
Bernshteyn first encountered descriptive set theory as an undergrad, where it was dismissed as a once-vital field that had faded into obscurity. It took over a year for him to learn that view was outdated.
During his first year of grad school in 2014 at the University of Illinois, he enrolled in a logic class taught by Anush Tserunyan (https://www.math.mcgill.ca/atserunyan/), who later served as his mentor. She set the record straight. "She deserves full props for drawing me into this area," he shared. "She portrayed logic and set theory as the essential binder holding diverse math branches together."
The roots of descriptive set theory trace back to Georg Cantor, who in 1874 established that infinities vary in scale (https://www.quantamagazine.org/how-can-some-infinities-be-bigger-than-others-20230419/). For instance, the collection of whole numbers (0, 1, 2, 3, and so forth) matches the scale of all fractions but falls short of the vast array of real numbers. To make this clearer for newcomers, picture the whole numbers as countable steps on a ladder, fractions as dense points on a number line between 0 and 1, and real numbers as every possible position on that line—infinitely more crowded.
Anush Tserunyan views descriptive set theory as the vital framework linking various mathematical domains.
Photograph: Courtesy of Anush Tserunyan
Back then, this assortment of infinities left mathematicians uneasy. "It's tough to get your mind around," admitted Bernshteyn, now based at the University of California, Los Angeles.
To ease that unease, they crafted an alternative way to gauge size—not by counting elements, but by assessing attributes like length, area, or volume. This is called a set's "measure," distinct from Cantor's "cardinality," which counts elements. A basic example is the Lebesgue measure, which measures length. The reals from 0 to 1 and from 0 to 10 both have infinite cardinality, but their Lebesgue measures are 1 and 10, respectively—think of it as the 'space' they occupy rather than the 'items' they hold.
Georg Cantor unveiled that infinity in math isn't uniform; it manifests in diverse forms and magnitudes.
Photograph: Emilio Segre Visual Archives
For trickier sets, other measures come into play. The more irregular the set, the scarcer the viable measurement methods. Descriptive set theorists investigate which sets qualify under various measure definitions, then rank them hierarchically. The upper tiers house simple, constructible sets amenable to any measure. The lower ones hold "unmeasurable" sets—so convoluted they're beyond quantification. "People often call them 'pathological,'" Bernshteyn explained. "These nonmeasurable sets are truly troublesome; they're intuitive opposites and don't play by expected rules."
This ranking isn't merely a map for set theorists; it equips them with strategies for broader math problems. Fields like dynamical systems, group theory, and probability often require set size insights, and a set's hierarchy spot dictates the available tools. For example, in probability, knowing if a set is measurable helps calculate event likelihoods accurately.
In essence, descriptive set theorists act as custodians of a grand library of infinite sets and their measurement methods. They assess a problem's set complexity, assign it to the right category, and alert other mathematicians accordingly.
Navigating Decisions
Bernshteyn is part of a subgroup organizing issues with infinite node-edge structures known as graphs. Specifically, he examines graphs with countless disconnected components, each brimming with endless nodes. Conventional graph experts usually stick to finite versions, but these infinite ones model dynamical systems and key set types, drawing keen interest from descriptive set theorists.
Consider a typical infinite graph Bernshteyn might analyze: Begin with a circle packed with infinite points. Select one as your initial node, then advance a set distance along the edge—say, one-fifth of the circumference—for the next node, linking them with an edge. Repeat: same distance to the third, connect it, and continue.
Advancing one-fifth each time loops back after five steps. Generally, fractional distances create cycles. But irrational distances yield an eternal chain of linked nodes. And that's just one segment. Since it misses most circle points, initiate new chains from remaining spots using the identical step size, yielding fully isolated infinite paths.
Repeat for every starting point, resulting in a graph of infinite disjoint infinite-node components.
Illustration: Mark Belan/Quanta Magazine
Now, pose queries like: Can nodes be colored under specific constraints? With two colors, can you ensure no adjacent nodes match? It appears simple: For each segment, start with blue, alternate yellow-blue, and replicate across all. Two colors suffice.
Photograph: Mark Belan/Quanta Magazine
But here's where it gets controversial—this relies on the axiom of choice, a foundational math tenet allowing selection of one element from each of infinite sets to form a new one. It's handy for proofs but spawns paradoxes, which descriptive set theorists sidestep. Your infinite segments equate to infinite sets; picking blue starters invokes it.
Alternating then scatters colors without inter-segment ties, rendering blue and yellow sets unmeasurable—no length insights possible. For set theorists, this feels incomplete; they seek choice-free, continuous colorings yielding measurable sets.
Recall segment construction: Node to spaced node. Color first blue, arc between blue; next yellow, arc yellow; alternate. Eventually, a tiny uncolored arc remains. If last was yellow, blue clashes with initial blue connections, yellow with prior yellow—necessitating green.
Illustration: Mark Belan/Quanta Magazine
Yet these color sets are contiguous arcs, measurable by length—far better than scattered points.
Thus, two-coloring lands low in the hierarchy (unmeasurable), three-coloring higher (broadly measurable). And this is the part most people miss: Does relying on the axiom of choice make math 'less pure,' or is it a necessary evil? It's a debate that riles purists.
Bernshteyn devoted grad school to classifying such colorings. Post-degree, he devised a unified shelving method, exposing their profound, underappreciated structure.
Layer by Layer
Occasionally, Bernshteyn attends computer science seminars, where graphs depict finite computer networks.
A 2019 presentation altered his path. It covered "distributed algorithms"—coordinated instructions across networked devices sans central control.
Envision Wi-Fi routers in a structure: Adjacent ones clash on shared channels, so each picks a unique one from neighbors.
This translates to graph coloring: Nodes for routers, edges for proximity; color so adjacents differ, representing channels.
Catch: Local algorithms only—nodes act on neighborhood info. Initially, self-color; share with neighbors; reassess and adjust iteratively till resolved.
Experts gauge steps needed. Two colors for routers? Inefficient. Three? Swift. For beginners, it's like neighbors negotiating paint colors without a boss, round-robin style.
The talk highlighted efficiency thresholds, one echoing descriptive set theory's color needs for measurable infinite graphs.
To Bernshteyn, it screamed deeper unity. Beyond shared graph-coloring motifs, perhaps their 'libraries' aligned perfectly, just in alien tongues—yearning for translation.
Unlocking Possibilities
Bernshteyn aimed to formalize this. He sought to equate efficient local algorithms with Lebesgue-measurable infinite graph colorings (plus key traits), linking pivotal shelves across fields.
Starting with the lecture's network issues, he emphasized locality: Algorithms draw solely from immediate surroundings, scaling from small to vast.
Success hinges on uniquely numbering neighborhood nodes for logging and directing. Finite graphs? Easy numbering.
The computer scientist Václav Rozhoň has leveraged this fresh set theory-network link to crack his favored puzzles.
Photograph: Tomáš Princ, Charles University
Extending to infinite graphs promised measurable coloring—but uncountable infinity defies unique labeling.
Bernshteyn's hurdle: Smart labeling with reuse, ensuring neighborhood uniqueness.
He proved it's feasible, regardless of labels or neighborhood size—enabling seamless algorithm transfer. "In our framework, any algorithm maps to a measurable coloring for descriptive set theory graphs," Rozhoň affirmed.
This stunned the math world, forging ties between computation, definability, algorithms, and measurability. Now, they're exploiting it: Rozhoň's team recently showed tree colorings via computer views (https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.29), aiding dynamical system tools. "Proving in an unfamiliar field, sans basics, is thrilling," Rozhoň quipped.
Reverse translations abound too—like set theory yielding complexity bounds (https://arxiv.org/abs/2111.03683).
Beyond tools, the bridge clarifies set theory's map, reclassifying enigmas using computer organization.
Bernshteyn envisions shifting perceptions: Set theory as integral, not aloof. "I'm pushing for infinity to become routine thinking."
But here's a counterpoint to ponder—is this bridge truly bridging 'pure' math with 'applied' tech, or just repackaging old ideas? Does it dilute math's abstract beauty? What do you think—does infinity deserve more spotlight in computer science, or should we keep worlds separate? Share your takes in the comments; I'd love to hear if you're team 'unite' or 'divide'!
Original story (https://www.quantamagazine.org/a-new-bridge-links-the-strange-math-of-infinity-to-computer-science-20251121/) reprinted with permission from Quanta Magazine (https://www.quantamagazine.org/), an editorially independent publication of the Simons Foundation (https://www.simonsfoundation.org/) whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.
