Hilbert's hotel

Example

A hotel has rooms numbered $1, 2, 3, \dots$, one for each positive integer, and each room has a guest. A new guest arrives. The manager moves the guest of room $n$ to room $n + 1$ for each $n$. Then room $1$ is free for the new guest. Next, infinitely many new guests $g_1, g_2, \dots$ arrive. The manager moves the guest of room $n$ to room $2n$, and gives room $2k - 1$ to $g_k$. In both cases, each guest has a room and no two guests share a room.

Remarks

The maps $n \mapsto n + 1$ and $n \mapsto 2n$ are injections of $\mathbb{N}_{>0}$ into a proper subset of itself. So the set of rooms is equinumerous with a proper subset. This cannot happen for a finite set. David Hilbert introduced the hotel in a lecture of 1924 to 1925, and George Gamow made it popular in a book of 1947.

Explanations

L1 human

A full finite hotel cannot take a new guest, but a full infinite hotel can. The example shows that "as many as" for infinite sets does not behave like counting.