Set theory

Zermelo-Fraenkel set theory is so widely accepted that modern mathematicians hardly think about it. But believing in its core principles didn’t come easily.

Ultrafinitism, a philosophy that rejects the infinite, has long been dismissed as mathematical heresy. But it is also producing new insights in math and beyond.

In an 1874 paper, Georg Cantor proved that there are different sizes of infinity and changed math forever. A trove of newly unearthed letters shows that it was also an act of plagiarism.

Intuition breaks down once we’re dealing with the endless. To begin with: Some infinities are bigger than others.

Researchers have used metamathematical techniques to show that certain theorems that look superficially distinct are in fact logically equivalent.

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.

Two new notions of infinity challenge a long-standing plan to define the mathematical universe.

A new proof illuminates the hidden patterns that emerge when addition becomes impossible.

A team of four prominent mathematicians, including two Fields medalists, proved a conjecture described as a “holy grail of additive combinatorics.”