Infinity

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.

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.

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.

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.

In math and computer science, researchers have long understood that some questions are fundamentally unanswerable. Now physicists are exploring how even ordinary physical systems put hard limits on what we can predict, even in principle.

Useful mathematical concepts, like the number line, can linger for millennia before they are rigorously defined.

Transcendental numbers include famous examples like e and π, but it took mathematicians centuries to understand them.