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.
In the late 19th century, mathematicians showed that most familiar infinite collections of numbers are the same size. This group includes the counting numbers (1, 2, 3, . . .), the even numbers, and the rational numbers (quotients of counting numbers, such as 3/4 and 101/763). However, in work that astonished the mathematicians of his day, the Russian-born Georg Cantor proved in 1873 that the real numbers (all the numbers that make up the number line) form a bigger infinity than the counting numbers do.