Welcome to Hilbert’s Grand Hotel: a Crash Course in Understanding Infinity

Welcome to Hilbert’s Grand Hotel: a Crash Course in Understanding Infinity

Did you know that the entire number line is the same size as the area between zero and one? Welcome to the strange world of infinity, where nothing is as it seems and sometimes the best response is none at all. David Hilbert, the famed mathematician who formulated the 23 questions that would define 20th-century math, devised one of the most famous thought experiments to help explain the bizarreness of infinite. Hilbert’s Grand Hotel is how it’s become known and it usually goes like this:

Imagine you’re on a trip, and you want to turn in for the night. You finally find a hotel – the only one for miles around – but alas, you see the sign says “no vacancies.” But look, you need somewhere to sleep, and the hotel looks pretty big, so you decide to go in and ask for a room anyway – just in case.

“We’re completely full,” says the check-in clerk, “not a single free room in the house.”

Disappointed, you turn to leave, but she stops you.

“Wait!” she says. “We can still fit you in – you see, this is a particularly special hotel. It has an infinite number of rooms. All we have to do is tell everybody already staying in the hotel to move into the next room over!”

She presses a button and speaks into the intercom.

“This is a customer announcement,” she says. “The guest staying in room one must please move into room two. The guest in room two is to move into room three. The guest in room three, please move to room four, and so on.” She turns back to you with a smile on her face.

“There we go,” she says. “Room one should now be free. I’ll check you in.”

As a result, you take your key and proceed to your room. Then you notice a swarm of tourists walking through the entrance.

“Hello,” says the group’s leader to the clerk. “We’re a group of 20 people, but we heard this place can always accommodate a few more.”

The cashier responds, “That’s right.” “Let me just move a few people around.”

She dials the intercom once more.

She says, “Another customer announcement.” “At the hotel, we have 20 new guests. Please walk to room 20 to your right: room one, room two, room three, room four, room five, room six, room seven, room eight, room nine, room ten, room eleven, room twelve, room thirteen, room fourteen, room fifteen, room sixteen, room seventeen, room eight Thank you very much!”

She returns her attention to the gathering.

She checks the group into rooms one through twenty and exclaims, “That should do it!” You have to admit, you’re impressed with this clerk: she effortlessly crammed 21 more customers into a completely full hotel. But suddenly the phone rings, and you notice her troubled expression.

“Are you sure? Yes, ma’am, we’ll see what we can do,” she adds before hanging up the phone.

She tells you, “There’s a full bus of travelers coming this way.” “There is an unlimited number of them, and they all need a bed for the night — we’ll have to increase the number of hotel rooms to accommodate them all!”