Guys, this is a public service announcement: infinity is CRAZY. Infinity is BIG. In fact it is SO BIG that when you’re talking about it, you TRIP over PARADOXES like they are LEGO pieces and BOXES in your house in the middle of the night when you just want to PEE.

One of the many paradoxes associated with infinity is Hilbert’s Paradox of the Grand Hotel *flourishes dramatically, throws sparkles in the air*. This grandiosely named conundrum takes place in a completely normal, everyday sort of building: a hotel with infinite rooms.

Ok, I lied about it being normal. Obviously, such a building could not exist. I mean, acquiring the building permits alone would be an impossible goal. But this is math, and one of the best things about math is that if you say an axiom is true, it might as well be, until you prove yourself wrong. Anything can happen! Except for the things that can’t! Don’t quote me on this! I am not a mathematician! That picture of my calculus notes is as far as my math education goes, and I’ve forgotten most of it. 

Back to the hotel with infinite rooms. As it just so happens, the hotel is pretty busy today. In fact, every single room is booked! Every numbered room has exactly one person staying in it. (I bet the person in room Googol felt pretty special, 42 was reserved early by a massive nerd, and whoever got the room with the smallest uninteresting number was really confused.)

So, the entire hotel is full. Then, a weary traveler shows up with his suitcase. Even though the hotel owner has infinite dollars now, from all these bookings, he wants to fit one more in. But he can’t, right? The hotel is full, and ‘full’ literally means ‘cannot fit any more’.  

But remember, this is infinity we’re talking about here. And as I mentioned, infinity is CRAZY. The hotel owner decides that it’s worth a little inconvenience on everyone’s part to get one more guest in, so here’s what he does: the person in room 1 is moved to room 2, that person is moved to room 3, the person in room 3 is moved to room 4, and so on and so forth.

Now, if you tried to do this with a finite amount of rooms, eventually you’d get to the end of the rooms, and someone would be kicked out. But there is no end to the rooms. They go on forever. There will always be another room to move the person to, no matter how far along you go. And so, with room 1 left vacant, the newcomer can slide right in!

This doesn’t just work for one person. For a family of five, you’d just move everyone up by five. For a group of one million, you’d simply ask everyone to walk past one million doors, and stay there instead. Easy!

But what if it’s not a group of one million? What if it’s a group of… infinity?

A HUGE bus carrying an infinite number of people drives up to the hotel. The hotel owner’s eyes are bugging out of his head. He could double his money! Of course, two times infinity is still infinity, but a savvy businessman would never turn such an offer down!

But how can the hotel owner fit these new guests? Your immediate guess might be that he moves everyone up an infinity of rooms, but of course, you can’t do that, because there is no end to infinity. They’d never reach their rooms, not in all of eternity, and that leads to bad Yelp reviews. That simply won’t do. So, here’s what the hotel owner does: the person in room 1 is moved to room 2. So far, it’s the same as before. But then, the person in room 2 is moved to room 4. The person in room 3 is moved to room 6. The person in room 10 is moved to room 20. Everyone is moved to the room with the number that is twice their current number. This is pretty easy for the person in room 25, but for the person in room 28,738,789,723,478,932, it’s a bit harder. Many- in fact, almost all of the people will need to walk or take the elevator for longer than the universe has existed. But if they are immortal, and very determined, they will, eventually, get there.

And the neat thing is, since everyone multiplied their number by 2, everyone is in an even-numbered room. This leaves all the odd-numbered rooms open, ready for the bus full of infinite people to check in!

I don’t think I’ll try to draw this into a larger context today. I think it’s amazing enough as it is.


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One thought on “Hilbert’s Paradox of the Grand Hotel

  1. Here’s another for you. Pick (but don’t “remove”) a number randomly, choosing from the numbers between 0 and 1. That is your target. It’s a number between 0 and 1. It exists and is a perfectly legitimate number. We know what it is now. It’s “target”.

    Now, do it again. What is the probability that you picked the same number?

    It has to be =possible=, since “target” is in the right range. The probability has to be the same as for any other number in the range, since those are all equally legitimate possibilities. The probability cannot be greater than zero, because that would lead to a total probability (of picking =anything=) greater than one. (It would lead to a total probability of infinity!)

    So, it has to be probability zero. But it can happen. Things with probability zero can happen!

    (It is not however true that things that cannot happen have a probability that is nonzero.)



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