Uncountable infinity

The set of real numbers is an uncountable infinity, larger than the countable infinity of the natural numbers. There are infinitely many different sizes of infinity, more than any one of them.

Cantor's Cruise Ship has pulled into the harbor in front of Hilbert's Grand Hotel, carrying a passenger for every real number. Every passenger has a ticket with a distinct real serial number, and every real number is realized—passenger √2, passenger e, passenger π and so on, all eager to check in to the famously accommodating hotel. Can the hotel manager fit them in?

The fascinating answer, shocking and profound, is No, the passengers on Cantor's cruise ship will not fit into Hilbert's hotel—there is no system of room assignments that will assign every passenger their own private room. In plain mathematical language, the set of real numbers cannot be placed into a one-to-one correspondence with a set of natural numbers. Thus Cantor proved that the set of real numbers is uncountable.

The fantastic conclusion is inescapable: there are different sizes of infinity. The infinity of the real numbers is a larger infinity than the infinity of the natural numbers. Indeed there are infinitely many different sizes of infinity, but no largest infinity. To be sure, there are more infinities than any particular one of them.

\( ℕ\quad < \quad ℝ\)

Although Cantor's ideas were controversial in his day, sometimes meeting with stubborn rejection, mathematicians eventually recognized his enormous accomplishment. In 1925, Hilbert extolled Cantor's theory of sets and infinite cardinals, announcing, “No one shall cast us from the paradise that Cantor has created for us.” Let us begin to see what Hilbert was talking about.