Hilbert's Hotel, A Thought Experiment on Infinity

Hilbert's Hotel is a thought experiment introduced by the German mathematician David Hilbert. It is used to illustrate the properties of infinite sets. The hotel has infinitely many rooms, numbered 1, 2, 3, and so on. Even when every room is occupied, the hotel can still accommodate new guests.

If a single new guest arrives, each current guest can be moved one room forward, freeing up room 1. If infinitely many new guests arrive, the existing guests can be moved to even-numbered rooms, making the odd-numbered rooms available for the newcomers. In a further extension, we can imagine infinitely many buses arriving, each carrying infinitely many guests. In that case, each guest can be assigned a unique room using a pairing function or by encoding their position using powers of prime numbers.

This thought experiment illustrates how infinite sets behave differently from finite ones. An infinite set that can be put into one-to-one correspondence with the natural numbers is called a countably infinite set, and its cardinality is denoted by $\aleph_0$ (aleph-null). Hilbert's Hotel demonstrates several basic results of cardinal arithmetic:

• $\aleph_0 + 1 = \aleph_0$: Adding one element to a countably infinite set does not change its size.
• $\aleph_0 + \aleph_0 = \aleph_0$: The union of two countably infinite sets is still countably infinite.
• $\aleph_0 \times \aleph_0 = \aleph_0$: The set of all ordered pairs of natural numbers is also countably infinite.

These results do not hold for finite sets, and they show that understanding infinite sets requires precise mathematical definitions rather than intuition. Hilbert’s Hotel is a helpful example for illustrating the unique structure of the infinite.