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 that 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 And infinite.
We can sum it up with a question: imagine you have a hotel with endless roomsall busy. If they start coming 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 we are Tonythe sympathetic receptionist who can solve all problems and who works in an endless hotel.
One day Tony is behind the counter in the lobby and is found with the hotel completely fulltherefore with an infinite and accounting number of guests occupying all endless rooms. The number of hotel rooms is equal to number of guests.
But then, during a working day, a new customer enters the hall and asks for a room. But wasn’t the hotel full? How you do it?
If you’re thinking “well, just put himself in the last room and we’re good to go”, know that no, he can’t. Since the rooms are infinite, there is no last “empty room”, they are all occupied.
Let’s see what the solution Finding: Tony asks the guest in the room number 1to move to 2, to the lady from 2 to move to 3, to the guest from 3 to move to 4 and so on for all the others. Each guest moves therefore from your room that we can call “n“To the room”n + 1“. As there are infinite rooms, there is a new one for each client.
In this way he manages to do free room number 1, where he can go to sleep on new client.
Here is the paradox: even now that a person has been added the number of rooms corresponds to number of guests that occupy them.
The same method applied to the new customer can be repeated for each finite number of new guests. 1,2,3,4,5, even 50 or 500 new guests.
Second level of difficulty
But we come to a more bizarre caselet’s move on 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 countless other guestsall who want to stay right in the Hotel.
At first, clearly, Tony gets very scared. Where is he going to 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 is asking each guest to move from the room of departure “N” at the room “nx 2”.
This method works very well because it does this can be filled the infinite ones rooms marked with even numbers (just doing nx2) and, at the same time, to leave free all the endless odd rooms! In doing so, the 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, so 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 accounting) of passengers!
Now how does Tony fix them all?
This time the solution is not own intuitive – not that in the other cases it was a walk in the park – but here we are at an even greater level of difficulty and to solve the riddle we have to use Prime numbers And powers. Let’s start with the guests who are already in the hotel.
Each of them must go to the room “2“ raised to their room number. Why 2? Because this method is based on prime numbers and 2 is the first prime number. That is, who is in the room 4 will go into room 24, then 16, who is in the room 10 will go into the room 210that is, in 1024.
Same thing for the passengers in the endless trains, they change only prime numbers which form the basis of power.
For example, whoever is on the first train will go to the room numbered with the next prime number (hence the 3) raised to its seat number (for example the 15). Then it will end up in the room 315 which is 14.348.907. Whoever is in the other trains will do the same thing but continuing with the other prime numbers, i.e. 5, 7, 11, 13, 17 etc. There will never be overlapping of rooms like this but, incredibly, there will even be empty rooms!
How do they stay empty rooms?
In fact, let’s take the case of … Ambrosea hotel guest, who at first was in the room 6 and – through the mechanism we have just now – it comes moved to room 26 that is the number 64. The room 6 However it will remain empty and will not be occupied by other patrons 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 will never get 6. So empty room.
Despite the emptiness left in some rooms, the owners of the Hotel certainly won’t get mad at Tony, his receptionist work was nevertheless exceptional.
You have indeed shown that too if the infinite hotel is all busy, it is always possible to host an infinite number of new customers.
There are various types of infinity
How is it possible that Hilbert’s absurd paradox has been resolved?
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 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 such as ⅔ or -¼), real ones with roots and complex numbers, which indeed are really super complex.
So when we talk about infinity we don’t have to think of just one thing. Exist many types of infinity And some they are infinite bigger than others!