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- Gedankenexperiment
- infinite set
Infinite Hotel Paradox
What is the Infinite Hotel Paradox?
How does the Infinite Hotel accommodate a new guest when it is full?
What is a countably infinite set?
What is uncountable infinity?
Imagine arriving at a hotel that claims to have infinitely many rooms: room 1, room 2, room 3, and so on forever. You walk to the front desk and say, “I need a room.” The manager looks up from her ledger and says, “Every room is taken—but don’t worry, I can make space.” She phones every guest and says,
“Please move to the room with the next highest number.”
So the guest in room 1 goes to room 2, the guest in room 2 goes to room 3, and so on.
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Since there is no “last” room, everyone shifts to the room with the next highest number, which frees up room 1 for you. This scenario—the Infinite Hotel Paradox, also called Hilbert’s Paradox of the Grand Hotel—illustrates a mathematical paradox showing how infinity behaves: A system can be completely full and yet make space for more.
The idea
The story was introduced by German mathematician David Hilbert in a 1924 lecture to explain how infinite sets differ from finite ones. The hotel has a room for every whole number: 1, 2, 3, 4, 5, and so on forever. Each guest can be designated by the guest’s current room number, n. Hilbert imagined the manager using a simple rule to move people around:.Here f(n) tells us the new room number for guest n. The guest in room 7 goes to room 8, the guest in room 10 goes to room 11, and so on. Everyone still has a room, but the first room is now empty. The size of the hotel is described as countably infinite, which is written as ℵ0 (aleph-null), which represents the smallest form of infinity. The result of this shift can be written as.This means that when you add 1—or any other finite number—to a countably infinite set, the set’s size doesn’t increase. Infinity doesn’t grow; it remains infinite.
A countably infinite set is one whose elements can be listed in order, even if the list never ends. Although such sets go on forever, their elements can still be counted one by one. Here are examples of countably infinite sets:
- All whole numbers (0, 1, 2, 3, 4,…)
- All even numbers (2, 4, 6, 8,…)
Making more space
Once you accept the rules of infinity, you understand that the hotel can handle any crowd.
A few extra guests
If 10 new people show up, the manager uses the rule.This means that she moves everyone 10 rooms forward. As a result, rooms 1 through 10 open up for the newcomers.
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A busload of infinite guests
If a bus arrives with infinitely many passengers, the manager tells each current guest to move to a room whose number is double the guest’s current room number:.The guest in room 1 moves to room 2, the one in room 2 moves to room 4, and so on. Consequently, every odd-numbered room—1, 3, 5, 7…—is freed up. Mathematically, this shows.This means that adding infinity to infinity still gives infinity.
Hilbert’s Hotel works for countable infinity. But there is a larger kind of infinity that cannot be arranged in a sequence. This is called uncountable infinity. A set is uncountably infinite when its elements are so densely packed that, no matter how you try, you can’t list them in a step-by-step order without leaving some out. The classic example is the set of real numbers—all the numbers on a number line. Between any two real numbers, no matter how close, there is always another number. You can’t count such numbers because you can’t start listing them without missing infinitely many in between.
Hilbert’s Hotel relies on moving to or relabeling rooms—pairing each guest with a room number. That’s possible when the infinity involved is countable. But if a new set of guests represented an uncountable infinity—such as every point on a line or every real number between 0 and 1—no such pairing would be possible.
When infinite buses arrive
Now imagine the hotel facing its biggest challenge yet: Not just one bus of new guests arrives, but infinitely many buses do—each carrying infinitely many passengers. At first, this situation may seem hopeless. But the manager has a plan. She labels every passenger by using two numbers:
- The bus number (bus 1, bus 2, bus 3,…)
- The seat number (seat 1, seat 2, seat 3,…)
So one person may be labeled “bus 2, seat 5,” another “bus 7, seat 1,” and so on. Thus, the total number of passengers would equal the number of all possible pairs of numbers—one for the bus and one for the seat. Mathematically, this means.To assign rooms, the manager imagines everyone arranged in a huge table:
| bus 1 | bus 2 | bus 3 | ... |
|---|---|---|---|
| (1, 1) | (2, 1) | (3, 1) | ... |
| (1, 2) | (2, 2) | (3, 2) | ... |
| (1, 3) | (2, 3) | (3, 3) | ... |
| ... | ... | ... | ... |
Every cell in the table represents one passenger. If the manager tried to check people in column by column, she would never get past counting the people from bus 1, as each row is infinite. So she changes the order of counting by going diagonally across the grid. She starts with the passengers whose bus and seat numbers add up to the same total:
- For (1, 1), the sum is 2
- For (1, 2), (2, 1), the sum is 3
- For (1, 3), (2, 2), (3, 1), the sum is 4, and so on forever
This diagonal counting lists everyone in a single never-ending line: Each person eventually gets a turn, so each gets a room. Mathematically, that means.It expresses that a countably infinite number of infinite groups can still be counted one by one.
The story behind the story
Hilbert introduced the Infinite Hotel during a lecture at the University of Göttingen in 1924 to illustrate the properties of infinity. His assistant Lothar Nordheim later recorded the example in Hilbert’s notes. Hilbert was building on the ideas of Georg Cantor, the founder of set theory, who was the first to define different sizes of infinity. Moreover, Hilbert believed that infinity exists only as a concept, not as something we encounter in nature. In his 1925 lecture “Über das Unendliche” (German: “On the Infinite”), he concluded,
The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought.…The role that remains for the infinite…to play is solely that of an idea…which transcends all experience and which completes the concrete as a totality.
More than two decades later, physicist George Gamow—who had worked at Göttingen in 1928 and later became known for his contributions to nuclear physics and early big bang theory—brought Hilbert’s thought experiment to a wide public. In his book One, Two, Three…Infinity: Facts and Speculations of Science (1947), Gamow recounted the story of the Infinite Hotel with humor, crediting it to Hilbert and joking in a footnote about an “unpublished, and even never written, but widely circulating volume titled ‘The Complete Collection of Hilbert Stories’ ” by Richard Courant, a former assistant of Hilbert’s. Gamow’s version of the paradox, mixing wit with clear explanation, helped turn Hilbert’s classroom illustration into a familiar example of how infinity defies everyday logic.