The Infinity Paradox of Hilbert’s Hotel

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Hilbert’s Hotel: The Hotel That Is Always Full but Never Out of Room

Hilbert’s Hotel is a thought experiment proposed by German mathematician David Hilbert in 1925. It involves a hotel with an infinite sequence of rooms: 1, 2, 3, and so on to ∞. This is called countable ∞, since each room can be associated with a counting number. The hotel starts out fully occupied.

One New Guest

A new person shows up and wants a room. Surprisingly, you can accommodate the new guest without removing any current guests. The hotel simply moves the guest in room 1 to room 2, the one in room 2 to room 3, and so on. In general, the guest in room N moves to room N + 1. Afterward, room 1 is free for the new guest.

Infinitely Many New Guests

Hilbert’s Hotel can also accept a countably infinite number of new guests. The guest in room 1 moves to room 2, the one in room 2 to room 4, and in general, the guest in room N moves to room 2N. This leaves all the odd-numbered rooms free for the new guests.

Hilbert’s Paradox

The fact that Hilbert’s Hotel can accommodate new guests despite being full is known as Hilbert’s paradox. This is an example of a veridical paradox, a statement that is true despite seeming false.

What Does This Tell Us About Infinity?

Hilbert’s Hotel illustrates one of the strangest properties of infinite sets: an infinite set can be put into a one-to-one correspondence with a proper subset of itself. The set of all natural numbers {1, 2, 3, …} has the same size as the set {2, 3, 4, …}, even though the second set seems to be missing an element. This is fundamentally different from how finite collections behave, where removing an element always makes the set smaller. Hilbert’s thought experiment makes this abstract property of countable infinity tangible and is one of the most widely used examples in introductory set theory for demonstrating that infinity is not just a big number but a concept with its own surprising rules.