Hilbert's Hotel

set-theory infinity paradox

Definition

Hilbert's Hotel

Hilbert’s Hotel is a thought experiment about a hotel with countably infinitely many rooms, indexed by , all of which are occupied.

Despite being full, the hotel can still accommodate additional guests by reassigning rooms using a bijection on the natural numbers.

The Shift

If the hotel is full and new guests arrive, one can move the guest in room to room for every .

This creates the rooms for the new guests while preserving a one-to-one correspondence between the original guests and the shifted rooms.

The shift map

is injective, and its image is the subset .

Why This Works

Although the hotel has no vacant rooms, the infinite collection of rooms can be reindexed so that finitely many rooms become available. This is impossible in a finite hotel, where moving every guest forward by one would require a room beyond the last room.