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 $1,2,3,\dots$, 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 $+k$ Shift

If the hotel is full and $k\in \mathbb{N}$ new guests arrive, one can move the guest in room $n$ to room $n+k$ for every $n\ge 1$.

This creates the rooms $1,2,\dots ,k$ for the new guests while preserving a one-to-one correspondence between the original guests and the shifted rooms.

The shift map

$${\sigma }_k:{\mathbb{N}}_{\ge 1}\to {\mathbb{N}}_{\ge 1},{\sigma }_k(n)=n+k$$

is injective, and its image is the subset $\{k+1,k+2,\dots \}$.

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.