## You are here

HomeHilbert's hotel

## Primary tabs

# Hilbert’s hotel

The hotel manager David Hilbert had a very large hotel, in fact, it had infinitely many rooms numbered $1$, $2$, $3$, $\ldots$. The hotel was very popular and every room was occupied. One day a new guest arrived.

-Is there any free room?

-No, Mr. Hilbert said.

-Oh, what a pity, the guest said and started to walk away.

-But wait, you can still get a room.

The new guest was very confused by this and asked how that could be possible.

-I’ll just ask the guest in room number 1 to move to room number 2, the guest in number 2 move to room 3, the guest in room 3 move to number 4, and so on, and then you can have room number 1.

The guest was very happy with this and called his friends to tell them about this fantastic hotel. Then one day they all arrived at the same time.

-Hello, we are countably many people, and we want a room each.
Mr. Hilbert felt reluctant to ask each guest to move to a new room an infinite number of times. That would be very unpleasant, and they will never finish either. Luckily he got a brilliant idea.

-I’ll let guest in number 1 move into number 2, guest in number 2 move into number 4, guest in number 3 move into number 6, number 4 to number 8 and so on.
Then you can move into any room with an odd number; they will all be free, and there will be rooms for all of you.

Now all the new guests got a room on their own.

Historical note:

David Hilbert is one of the great mathematicians in the history of mathematics. One of the things he studied was the foundations of mathematics and that includes the nature of infinity.

## Mathematics Subject Classification

03E10*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: Prove that for any sets A, B, and C, An(BUC)=(AnB)U(AnC) by St_Louis

Apr 20

new image: information-theoretic-distributed-measurement-dds.png by rspuzio

new image: information-theoretic-distributed-measurement-4.2 by rspuzio

new image: information-theoretic-distributed-measurement-4.1 by rspuzio

new image: information-theoretic-distributed-measurement-3.2 by rspuzio

new image: information-theoretic-distributed-measurement-3.1 by rspuzio

new image: information-theoretic-distributed-measurement-2.1 by rspuzio

Apr 19

new collection: On the Information-Theoretic Structure of Distributed Measurements by rspuzio

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

## Corrections

Linking by aoh45 ✓

typo and latex by matte ✓

spelling by Mathprof ✓

link suppess by Mathprof ✓