Hilbert’s infinite room hotel is a famous thought experiment in mathematics that helps us understand the weirdness of infinity. The German mathematician David Hilbert came up with it to show how infinite sets don’t follow the same rules as things we can count. 🏨
The basic idea is that a hotel with an infinite number of rooms, even if it’s completely full, can always make room for more guests.
The Hotel Setup
Imagine a hotel with an infinite number of rooms, numbered 1, 2, 3, 4, and so on, forever. One night, the hotel is totally booked—every single room is occupied. There’s a “No Vacancy” sign hanging out front.
Scenario 1: One New Guest Arrives
A traveler arrives late at night and asks for a room. Even though the hotel is full, the manager says, “No problem!”
- The Solution: The manager gets on the intercom and asks every guest to please move from their current room to the next-highest-numbered room.
- The guest in room 1 moves to room 2.
- The guest in room 2 moves to room 3.
- This continues for all of the infinite guests.
- The Result: Since every guest moves down one room, Room 1 is now empty. The new guest can move right in.
Scenario 2: A Bus with Infinite Guests Arrives
Just when the manager thinks the night is over, a bus pulls up carrying an infinite number of new passengers, all needing a room. The hotel is full again. What now?
- The Solution: The manager has another clever plan. This time, she asks every current guest to move from their current room number to the room number that is double their current one.
- The guest in room 1 moves to room 2 (1 times 2).
- The guest in room 2 moves to room 4 (2 times 2).
- The guest in room 3 moves to room 6 (3 times 2).
- The Result: All the current guests are now in the even-numbered rooms (2, 4, 6, 8, and so on). This means that all the odd-numbered rooms (1, 3, 5, 7, and so on) are now free. Since there are an infinite number of odd numbers, the hotel has just made an infinite number of vacancies for the new guests from the bus.
So, What’s the Point? 🤔
Hilbert’s Hotel isn’t a real place; it’s a story to show that infinity is not just a very large number. It behaves in completely different ways.
The paradox demonstrates that an infinite collection of things can be matched up with a smaller part of itself. For example, the set of all counting numbers (1, 2, 3, …) is the same “size” as the set of all even numbers (2, 4, 6, …), even though one is clearly a subset of the other. This is a key feature of infinite sets that just doesn’t work for finite things. If a hotel with 100 rooms is full, you can’t possibly fit one more person in. With infinity, you can fit infinitely more.