Hilbert’s Paradox Explained: The Grand Hotel with Infinite Rooms


Are you sure you know what it means”infinite“? Today we try to challenge you making you do an experiment.
Don’t worry, there is nothing risky because it is a thought experimenta test we can do using only our imagination: the Paradox of the Grand Hotel with infinite rooms. This experiment was developed by the mathematician David Hilbert in 1924 to show some features of the infinity concept and the differences between types of operations carried out with finite sets Y infinite.
We can summarize it with a question: imagine that you have a hotel with infinite bedroomseverybody occupied. if they start to come New customersis it possible to host them?
The answer could be much more complex than it seems! Let’s see it together.

The exposition of the paradox, first level

we imagine that we are toniumthe nice one receptionist who can solve all the problems and who works in an endless hotel.
One day, Tony is behind the lobby counter and is found with the hotel completely fulltherefore with an infinite and countable number of guests occupying all endless rooms. the number of hotel rooms is equal to number of invites.

hotel receptionist

But then, during a day at work, a new client He walks into the hall and asks for a room. But wasn’t the hotel full? How do you do it?
If you’re thinking “well, just go to the last room and we’re good to go”, know that no, it can’t. Since the rooms are infinite, there is no last “empty room”, they are all occupied.
let’s see what solution Finding: Tony asks the guest at the room number 1to go to 2, the lady from 2 to go to 3, the guest from 3 to go to 4 and so on for all the others. Every guest moves so from your room we can call “north“To the room”north + 1“. Since there are infinite rooms, there is a new one for each client.
In this way he manages to spare room number 1, where can you go to sleep new client.


Here is the paradox: even now that a person has been added the number of rooms corresponds to number of invites that occupy them.
The same method applied to the new client can be repeated for each finite number of new hosts. 1,2,3,4,5, even 50 or 500 new guests.

Second level of difficulty

But we come to a strangest caseLet’s go to the next level of difficulty.
One day Tony sees arriving in front of the hotel an endless train, very long! This endless train carries with it many other guestsall those who want to stay in the Hotel.
At first, Tony is clearly very scared. Where will he put all these people?


Here, too, our receptionist finds one solution: The trick is to ask the guest in room 1 to move to room 2, the guest in room 2 to move to room 4, the guest in room 3 to move to room 6. Basically, Tony asks each guest to Move on from the room of departure “NORTH” in the room “nx 2”.
This method works great because it does this can be filled the infinities rooms marked with even numbers (simply doing nx2) and at the same time to exit free everybody infinite strange rooms! By doing so, infinite new guests will be able to stay in infinite rooms, all marked with odd numbers.

The third level of the paradox.

But with all this success, many people want to stay at Tony’s Hotel!
Word spreads and then one day they show up in front of the hotel. endless trains with a infinite number (but always countable) of passengers!
Now how does Tony fix them all?
This time the the solution is not own self intuitive – not that in the other cases it was a walk in the park – but here we are at an even higher level of difficulty and to solve the puzzle we have to use Prime numbers Y powers. Let’s start with the guests who are already in the hotel.


Each of them must go to the room “two elevated to your room number. Why 2? Since this method is based on prime numbers and 2 is the first prime number. I mean, who’s in the room 4 enter to room 24, then 16, which is in the room 10 will enter the room two10that is, in 1024.
The same for the passengers in the endless trains, They change only Prime numbers that constitute the basis of power.


For example, whoever is on the first train will go to the room numbered with the next prime number (Therefore, the 3) raised to your seat number (For example, him fifteen). Then it will end up in the room. 3fifteen which is 14,348,907. Whoever is in the other trains will do the same but continue with the other prime numbers, that is, 5, 7, 11, 13, 17 etc. There will never be overlapping rooms. so but, incredibly, there will even be empty rooms!

How are the empty rooms?

In fact, let’s take the case of… ambrosea hotel guest, who first I was in the room 6 and – through the mechanism that we just had – comes moved into room 26 that’s number 64. room 6 Nevertheless will remain empty and it will not be occupied by other users because 6 is not a power of a prime number! That is, if I take any prime number and try to raise it to any other number, I’ll never get 6. So empty room.


Despite the emptiness that was left in some rooms, the owners of the Hotel certainly won’t be mad at Tony, however his work as a receptionist was exceptional.
In fact, you have also shown it. if the infinity hotel is all occupied, it is always possible to host an infinite number of new clients.

There are various types of infinity.

How is it possible that Hilbert’s absurd paradox has been solved?
Simple, the receptionist Tony has always had to deal with natural numbers (1,2,3,4,5,6 etc) which are accountingthat is, in mathematical terms, a discreet together.
Think about what natural numbers are. one of the simplest types of infinity theorized by mathematics!
In fact, this infinite set does not include negative integers (-1, -2, -3, etc.), rational numbers (those attributable to fractions like ⅔ or -¼), reals with roots, and complex numbers, yes they are really super complex.

So when we talk about infinity we don’t have to think of just one thing. Exist many kinds of infinity Y some they are infinite bigger than others!

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