A new approach to one of mathematics' most notorious problems
How many numbers are there? For children, the answer might be a million—that is, until they discover a billion, or a trillion, or a googol. Then, maybe they notice that a googol plus one is also a number, and they realize that although the names for numbers run out, the numbers themselves never do. Yet to mathematicians, the idea that there are infinitely many numbers is just the beginning of an answer. Counterintuitive as it seems, there are many infinities—infinitely many, in fact. And some are bigger than others.