Welcome to the Hilbert's Hotel! This is a hotel in Hilbert's imagination. There are infinite rooms and infinite customers. No matter how many guests, you have to set enough rooms to let them move in. Are you ready to be a manager?

background:

Alex is a manager of the Hilbert's Hotel.

To make a lot of money, Alex always works hard and tries to put every guests into the room in Hilbert's Hotel.

Hilbert's Hotel is a hotel with an infinite number of rooms. But there is a problem: the rooms in Hilbert's Hotel are all full.

## Condition1:

there is a finite number of guests want to rent the rooms

What should Alex do?

Alex put guest in room 1 move into room 2, guest in room 2 move into room 3, and so on. That is, move guests from room n into room n+1. Now the new guest can move into room 1!

the room 1 is empty!

The process can be repeated for any finite number of new guests.

(If there are x guests want to rent the room, just to move the guest in room n into room n+x.)

## Condition2:

there is an infinitely large bus with a countably infinite number of passengers

What should Alex do?

Alex put guest in room 1 into room 2, put guest in room 2 into room 4, and so on. That is, move guests from rooms n into room 2n.

Now there are infinitely many odd-number rooms, so every guest in the bus can get a room.

The room with odd number is empty!

## Condition3:

There is an infinite line of infinitely large buses each with a countably infinite number of passengers.

What should Alex do?

At first, Alex number all the buses and seats on each bus in ordered.

**In the hotel:** Then, Alex assigns every current guest to the first prime, 2, raised to the power of their current room number. That is, the guest in room n move into room \(2^n\).

**1st bus:** Assign the guests to the room number of the next prime, 3, raised to the power of their seat number on the bus. That is, the guest in seat n will move into room \(3^n\).

**2nd bus: **Assign the guests to the room number of the next prime, 5, raised to the power of their seat number on the bus. That is, the guest in seat n will move into room \(5^n\).

**3rd bus: **Assign the guests to the room number of the next prime, 7, raised to the power of their seat number on the bus. That is, the guest in seat n will move into room \(7^n\).

……

assign the room by prime number

Then repeat the process to each bus so that every guests could move into their room.

## Hilbert's paradox of the Grand Hotel

Hilbert's paradox of the Grand Hotel is a thought experiment. The idea was introduced by David Hilbert, a Geman mathematicain. This paradox is to show how hard it is wrap our minds around the concept of infinity. You may notice that the example above all talk about "**countable**" infinity. What if we talk about "**uncountable**" cases? Isn't that more difficult?

If you don't understand very well, you can watch this video The Infinite Hotel Paradox – Jeff Dekofsky on TEDEd.

##### Add some ingredients

♦ The Infinite Hotel Paradox – Jeff Dekofsky

♦ Hilbert's paradox of the Grand Hotel

##### Seasoned with Chinese

◊ The Infinite Hotel Paradox – Hilbert Hotel Paradox

◊ countable – countable

◊ uncountable – uncountable

◊ infinite – unlimited

◊ finite – limited