A question about infinity that math itself cannot answer
There are infinitely many whole numbers, and a strictly bigger infinity of real numbers. Georg Cantor asked in the 1870s: is there any set whose size sits strictly between those two infinities, or does every infinite subset of the reals match one of those two sizes exactly?
Reveal the answer
The astonishing resolution is that the question can't be settled using standard mathematical axioms: Kurt Gödel showed in 1940 that the continuum hypothesis can't be disproved from those axioms, and Paul Cohen showed in 1963 that it can't be proved from them either. It is independent of the foundations of mathematics -- true in some consistent versions of set theory and false in others.