Hilbert's Hotel
Overview
Imagine a hotel with infinitely many rooms, all of them occupied. A new guest arrives. The manager asks every guest to move to the room one number higher, and the new guest takes room 1. Infinitely many new guests arrive. The manager asks every guest to move to the room with double their number, freeing all the odd-numbered rooms. The hotel is always full, and always has room.
Hilbert's Hotel is a thought experiment devised by the mathematician David Hilbert in 1924 to illustrate the counter-intuitive properties of infinite sets: an infinite set can be put into one-to-one correspondence with a proper part of itself.
Notes
- William Lane Craig uses the hotel in the Kalam Cosmological Argument to argue that an actual infinite cannot exist in reality, and therefore that the past must be finite and the universe must have begun. The hotel is meant to show that actual infinities produce absurdities.
- It shows no such thing. The hotel is counter-intuitive, not contradictory. Mathematics handles infinite sets without inconsistency (Georg Cantor's work on transfinite numbers predates Hilbert's hotel by forty years), and the "absurdities" are only absurd by the standards of finite arithmetic, which is exactly what one should expect. That I find infinity strange tells you about my intuitions, which evolved to count sheep, not about reality. See Owens Tryptic of Ignorance and Personal Incredulity.
- Whether the physical universe contains actual infinities is an open question in physics, not one that can be settled by a story about a hotel.