Descriptive set theorists study the niche mathematics of infinity. Now, they’ve shown that their problems can be rewritten in the concrete language of algorithms.

All of modern mathematics is built on the foundation of set theory, the study of how to organize abstract collections of objects. But in general, research mathematicians don’t need to think about it when they’re solving their problems. They can take it for granted that sets behave the way they’d expect, and carry on with their work.

Descriptive set theorists are an exception. This small community of mathematicians never stopped studying the fundamental nature of sets — particularly the strange infinite ones that other mathematicians ignore.

Their field just got a lot less lonely. In 2023, a mathematician named Anton Bernshteyn(opens a new tab) published a deep and surprising connection(opens a new tab) between the remote mathematical frontier of descriptive set theory and modern computer science.

He showed that all problems about certain kinds of infinite sets can be rewritten as problems about how networks of computers communicate. The bridge connecting the disciplines surprised researchers on both sides. Set theorists use the language of logic, computer scientists the language of algorithms. Set theory deals with the infinite, computer science with the finite. There’s no reason why their problems should be related, much less equivalent.

“This is something really weird,” said Václav Rozhoň(opens a new tab), a computer scientist at Charles University in Prague. “Like, you are not supposed to have this.”

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