Continuing down the list, this mathematical method (called "diagonalization") generates a real number p between zero and one that, by its construction, differs from every real number on the list in at least one decimal place. Ergo, it cannot be on the list.

In other words, p is a real number without a natural number partner—an apple without an orange. Thus, the one-to-one correspondence between the reals and the naturals fails, as there are simply too many reals—they are "uncountably" numerous—making real infinity somehow larger than natural infinity.

"The idea of being 'larger than' was really a breakthrough," says Stanley Burris, professor emeritus of mathematics at the University of Waterloo in Ontario. "You had this basic arithmetic of infinity, but no one had thought of classifying within infinity—it was just kind of a single object before that."

Adds mathematician Joseph Mileti of Dartmouth College: "When I first heard the result and first saw it, it was definitely something that knocked me over. It's one of those results that's short and sweet and really, really surprising."