In the 1995 Pixar film *Toy Story,* the gung ho space action figure Buzz Lightyear tirelessly incants his catchphrase: "To infinity … and beyond!" The joke, of course, is rooted in the perfectly reasonable assumption that infinity is the unsurpassable absolute—that there is no beyond. That assumption, however, is not entirely sound. As German mathematician Georg Cantor demonstrated in the late 19th century, there exists a variety of infinities—and some are simply larger than others.

Take, for instance, the so-called natural numbers: 1, 2, 3 and so on. These numbers are unbounded, and so the collection, or set, of all the natural numbers is infinite in size. But just how infinite is it? Cantor used an elegant argument to show that the naturals, although infinitely numerous, are actually less numerous than another common family of numbers, the "reals." (This set comprises all numbers that can be represented as a decimal, even if that decimal representation is infinite in length. Hence, 27 is a real number, as is π, or 3.14159….)

*n*thus has a real partner

*r*. The reals can then be listed in order of their corresponding naturals:

_{n}*r*and so on.

_{1}, r_{2}, r_{3},*p,*by the following rule: make the digit

*n*places after the decimal point in

*p*something other than the digit in that same decimal place in

*r*. A simple binary method would be: choose 0 when the digit in question is 1; otherwise, choose 1.

_{n}*r*) is the decimal component of π (0.14159…), the pair for 2 (

_{1}*r*) is George W. Bush’s share of the popular vote in 2000 (0.47868…) and that of 3 (

_{2}*r*) is Ted Williams’s famed .400 batting average from 1941 (0.40570…).

_{3}*p*following Cantor’s construction: the digit in the first decimal place should not be equal to that in the first decimal place of

*r*which is 1. Therefore, choose 0, and

_{1},*p*begins 0.0…. Then choose the digit in the second decimal place of

*p*so that it does not equal that of the second decimal place of

*r*which is 7 (choose 1;

_{2},*p*= 0.01…). Finally, choose the digit in the third decimal place of

*p*so that it does not equal that of the corresponding decimal place of

*r*which is 5 (choose 1 again;

_{3},*p*= 0.011…).

*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.

*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."