Friday, 4 May 2007

Hilbert's Hotel


I've written about a few eccentrics on this blog and David Hilbert is another one. Hilbert was a German mathematician and geometrician and heavily involved in number theory. One of his students committed suicide after struggling with a problem set by Hilbert and he was invited to speak at the student's funeral. At the side of the grave he addressed the crowd and explained that the problem he set was actually quite simple, the student simply looked at it the wrong way. Hilbert's Hotel or Hotel Infinity was an illustration created by the mathematician to highlight the problems created by dealing with infinity as a number and I shall try and describe it to the best of my abilities.

Imagine, though don't worry if you can't, an infinite hotel with an infinite number of rooms numbered 1,2,3,4... and so on ad infinitum, and when you get there to check-in you find out that it's full despite the fact that the neon 'rooms available' sign is flashing outside. So you call for the manager and he reassures you that despite being full, he can still accommodate you all that is required is that the guest in room 1 moves to room 2, the guest from room 2 moves to room 3 and so on leaving everyone with a room and room 1 vacant for you. An infinite hotel is quite impressive so using the wireless internet in your room you log onto myspace or facebook and spread the word. The next day a thousand people arrive at reception eager to see this amazing hotel and again the manager has no problem in fitting everyone in. Everyone is shifted a thousand rooms, so you in room 1 are now moved into room 1001.

The next day an unexpected party arrives, an infinite number of holiday makers stop off at the hotel on their way to Legoland Windsor, it is frightfully popular. To fit this rather large number of new guests, the manager moves the person from room 1 to room 2, room 2 to room 4 and room 3 to room six and so on leaving an infinite amount of odd numbered rooms available for the infinite number of new arrivals. Now can you imagine staying at an infinite hotel, the queues for the lifts are, well infinite, room service is lamentable and you are sharing the bandwidth of the wireless broadband with an infinite number of other computers so you can appreciate that many guests are rather disgruntled and the next day all the guests in the even numbered rooms pack their bags and leave though despite going down to 50% capacity, the hotel still has infinite guests.

The last twist of the plot is that the chain who own the hotel close down an infinite number of infinite hotels and send all their guests to the Hotel infinity for accommodation. The manager becomes desperate for a solution, how do you accommodate an infinite number of infinite numbers of guests? So he asks his guests for some help, since you've observed everything that has gone on from the beginning you propose a solution. Remembering that you've only got people staying in the odd numbered rooms you move them into the even rooms, then the first person from the first hotel goes into the first empty room, room 1, the second person from the first hotel and the first person from the second hotel get the next two empty rooms, rooms 3 and 5 and the third person from the first hotel, the second person from the second hotel and the first person from the third hotel get the next three empty rooms and so on until everyone has a room.

Despite having 100% capacity in an infinite hotel with infinite turnover, costs are infinite and despite the accountant's good work at securing a low rate of tax, the liability is still infinite. The manager brings in his accountant to explain the situation and comes away reassured because even paying infinite costs and an infinite tax bills, he'll still be left over with infinite profits. David Hilbert's illustration leaves you with the question of whether there is an infinity which is bigger than another infinity. The answer to this question came from another German mathematician, Georg Cantor whose diagonal theorem apparently proves the existence of larger infinities but you can research that one for yourself. The Open University did a film of Hilbert's Hotel with Susannah Doyle (the scary black haired one from 'Drop the Dead Donkey') as the lead and it is well worth watching if you ever get the chance.

3 comments:

gary thomson said...

I don't know about eccentric, it sounds more emotionally retarded to point out the student's mistake at his graveside.

In a way it reminds me of when it came to light that the Unabomber was Ted Kaczynski and the papers were full of lines like evil genius refelecting the shock that the perpetrator of these crimes (murder after all) was a former maths teacher. One of Kaczynski's own former teachers, a Professor at Uni of California, was asked about him: He was a very brilliant student who solved a difficult problem in... (whatever it was) as though that was all that was important to know about the Unabomber.

You might be interested in the Banach-Tarski paradox.

...a marble could be cut up into finitely many pieces and reassembled into a planet, or a telephone could be transformed into a water lily.

Paolo said...

That is one of the aspects of infinity that I struggle to grasp, which is that something can not only be infinitely large but can also be infinitely small because everything is infinitely sub-divisible. It reminds me of that poem by William Blake:

"To see the World in a Grain of Sand,
And Heaven in a Wild Flower,
Hold Infinity in the Palm of your Hand,
And Eternity in an Hour"

The Banach-Tarski paradox is quite fascinating and it makes you think. The first representations of of the infinite came around 1,500 years B.C. with 'Ouroborous' and Zeno wrote about the paradoxes of the infinite at least 500 years B.C. so the scribes who wrote the appropriate sections of the New Testament dealing with Jesus and the fish would, in all probability, have known about it all in their role as scholars. I wonder whether you could interpret it solely as the illustration of a mathematical concept.

Mayda said...

Thanks for writing this.