Hilbert's Paradox of the Grand Hotel
Stimulating critical thinking - a loose series on paradoxes
Gödel's is a newsletter about interweaving ideas and making decisions under uncertain conditions. I discuss knowledge management, mental models, and supporting Tools for Thought.
Conceived by the German mathematician David Hilbert in 1924, the paradox is a thought experiment that illustrates infinity's strange and counterintuitive properties.
Imagine a hotel with a peculiar twist: it has infinite rooms. Now, imagine that every single one of these rooms is occupied. At first thought, you might assume that there's no way the hotel could accommodate any new guests.
Suppose a new guest arrives at the hotel. The hotel manager implements a simple but ingenious strategy: they ask the guest in room 1 to move to room 2, the guest in room 2 to room 3, and so on. In this way, each guest moves from room number n to room number n+1. Since the hotel has infinite rooms, there's always the next room to move into. Consequently, room 1 becomes available for the new guest despite the hotel being full.
The paradox becomes even more intriguing when we consider an infinite number of new guests arriving. In this scenario, the hotel manager asks each current guest to move from room number n to room number 2n (doubling their room number). This action frees up all the odd-numbered rooms, creating infinite vacancies for the new guests.
If an infinite number of buses, each with an endless number of guests, drive up, these guests can also be accommodated in the already full hotel. This can be done by freeing up the rooms with odd numbers as described above and then sending the guests from bus 1 to rooms 3, 9, 27, ... (i.e., to rooms numbered with powers of 3), the guests from bus 2 to rooms 5, 25, 125, … (powers of 5), etc. with using odd prime numbers as a base. This means that all guests who have arrived are accommodated in the hotel, and even an infinite number of rooms (such as room 15, whose number is not a power of a prime number) are still free.
The Grand Hotel may not be where you can book a room, but it's certainly a destination for the mind, offering endless rooms for thought and exploration.