Set theory

One of the strangest results in mathematics explains how it’s possible to turn one sphere into two identical copies, simply by rearranging its pieces.

A pair of researchers has shown that trying to classify groups of numbers called “torsion-free abelian groups” is as hard as it can possibly be.

For 50 years, mathematicians have believed that the total number of real numbers is unknowable. A new proof suggests otherwise.

His incompleteness theorems destroyed the search for a mathematical theory of everything. Nearly a century later, we’re still coming to grips with the consequences.

Explore our surprisingly simple, absurdly ambitious and necessarily incomplete guide to the boundless mathematical universe.

Two monumental works have led many mathematicians to avoid the equal sign. The process has not always gone smoothly.

Two mathematicians have proved that two different infinities are equal in size, settling a long-standing question. Their proof rests on a surprising link between the sizes of infinities and the complexity of mathematical theories.

A surprising new proof is helping to connect the mathematics of infinity to the physical world.

The Fields medalist Vladimir Voevodsky has died at 51. This 2015 article describes his computer-aided quest to eliminate human error and rewrite the century-old rules underlying all of mathematics.