INFINITY is a concept that is easy to think about, but hard to understand. Who hasn’t looked up at the night sky and wondered whether space goes on forever? Is it an endless expanse, or does it eventually just stop? What does it mean if it doesn’t?

For trained mathematical brains, the infinite is if anything even more bamboozling. Mathematicians have known for well over a century now that infinity isn’t just one thing, it is infinitely many. There is an unending tower of ever greater infinities stretching up all the way to… well, whatever you’d like to call it.

That isn’t even the worst of it. Although the existence of this tower of infinities is a logical consequence of mathematics as we know it, that same mathematics is powerless to describe it completely. Chip away at the plaster to reveal the structure underneath and you see that crucial load-bearing beams are missing in the lower levels, suggesting that the foundations of mathematics itself are unstable.

Mathematicians have long argued about how best to shore the infinite tower up. Some say we should simply leave well alone and hope for the best. Others have proposed fixes, variously deemed too costly, unlikely to work or not in keeping with the original style. No one has yet made anything like a breakthrough. Except, perhaps, until now. After decades of apparent stalemate, serious progress seems to have been made on the baffling question that lies at the heart of it all: a nearly 150-year-old unproven conjecture known as the continuum hypothesis.

Humans have probably been thinking about the unending for most of …