Infinity makes sense as a concept, even if we can’t necessarily wrap our heads around it. But according to some scholars, that’s not enough of a reason to believe in it. Welcome to the world of the ultrafinists—mathematicians who refuse to accept infinity unless it can be proven in a practical sense, not just a theoretical one. If you’re looking to break your brain on a Thursday, look no further than Gregory Barber’s piece about how far down those turtles actually go.
These physics-based challenges to the infinite tend to delight ultrafinitist mathematicians, who hold them up as evidence that their mathematics is a truer description of reality. At the 2025 conference, [Sean] Carroll’s talk on whether the universe is truly infinite or “merely quite large,” as he put it, made him something of a celebrity in the Columbia University halls. But the burden of proof, he cautions, remains with the infinity doubters. If you could somehow prove experimentally that the physical universe is indeed finite, even the most ardent backers of the higher infinite would likely take a moment to pause and reflect. They would probably even wonder about the consistency of set theory, given the towers of actual infinities that it allows. That’s a healthy thing to do from time to time, anyway.
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