INFINITY is a concept that is easy to think about, but difficult to understand. Who has not looked up at the night sky and wondered if space is eternal? Is it an endless stretch, or does it eventually just stop? What does it mean if it isn’t?
For trained mathematical brains, infinity is, if anything, even more deceptive. Mathematicians have known for over a century that infinity is not just one thing, it is infinitely many. There’s an endless tower of ever-increasing infinities that stretches up to… well, whatever you want to call it.
That’s not even the worst. Although the existence of this tower of infinities is a logical consequence of mathematics as we know it, that same mathematics is powerless to fully describe it. Remove the plaster to reveal the structure below and you’ll see crucial load-bearing beams missing from the lower levels, suggesting that the foundations of the math itself are shaky.
Mathematicians have long argued about the best way to prop up the infinite tower. Some say we should just leave it all alone and hope for the best. Others have proposed fixes, which are considered too expensive, unlikely to work, or not in keeping with the original style. No one has yet made anything resembling a breakthrough. Except, perhaps, until now. After decades of apparent stagnation, serious progress appears to have been made on the perplexing question at the center of it all: a nearly 150-year-old unproven conjecture known as the continuum hypothesis.
Humans have probably been thinking about the endless for most of…