---
tag: 0N3Y8G340X
kind: example
name: Hilbert's hotel
aliases: [hilbert-grand-hotel, hilberts-hotel]
msc: ["03E10"]
instance_of: [00CR00C9DN]
uses: [0SEZ3JGZP7, 0PVG1CRAJN]
sources:
  - url: https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel
    locator: lead section
same_as:
  wikidata: Q828646
  wikipedia: Hilbert's paradox of the Grand 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|0PVG1CRAJN]] 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|0BBTH3YSZA]].
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.