- Even with the advent of Calculus,
infinity itself in Mathematics remained poorly understood.
It was only in the late 19th century
that new mind-bending ideas helped tame
that strange beast, infinity.
(bell rings)
When I asked my friend,
author and mathematician Eugenia Chang,
to discuss her thoughts on infinity,
she suggested that we visit the imaginary Hilbert's Hotel.
A thought experiment,
first proposed by mathematician David Hilbert in the 1920s,
to demonstrate some of the odd properties of infinity.
And this hotel is definitely an odd property.
- Well, the Hilbert Hotel is a pretty amazing hotel.
It has an infinite number of rooms.
Wouldn't that be great?
You might think that you could always fit more people in,
but what if an infinite number of people showed up?
And then the hotel would be full.
- [Talithia] Oh dear.
- Then if another person came along, what would you do?
Well, if you weren't very astute,
then you might just say, "Sorry, we're full."
- That's one solution.
Or you might think,
given there are an infinite number of rooms,
you can just assign the late guest the room that comes
after the one given to the last guest that checked in.
You know, just farther down the hall.
Just put this person at the end of the line.
Why can't we do that?
- Where is the end of the line?
Sounds like a philosophical question,
but the thing is you can't just tell 'em to go to the end.
You have to give them a room number,
and all the rooms are full.
- [Talithia] Hmm. Seems unsolvable.
But luckily, any manager of a hotel
with an infinite number of rooms
and an infinite number of guests has
to have an infinite number of tricks up their sleeve.
- [Eugenia] Okay, how about the person in room one moves
into room two,
and the person in room two moves into room three,
and the person in room three moves into room four,
and so on.
Everybody has another room they can move into,
because everyone just adds one to their room number.
And that will leave room one empty.
- So new person comes, welcome.
You know what? We're just gonna have
everybody scoot over for you.
Just scoot, goes in room one.
- And then what if two people showed up?
That's fine. Everyone moves up two rooms.
What if five people show up?
That's fine.
- [Talithia] But what if an infinite number showed up?
Say, because of a fire at a second nearby
completely full Hilbert's hotel.
Is there room for a second infinity of guests?
- You've now got an infinite number of people.
You can't just get everyone to move up an infinite number
of rooms because where would they go?
- [Talithia] There is a solution.
The manager asks each person checked into a room
to multiply their room number by two and move there.
So one goes to two,
two goes to four,
three goes to six,
and so on.
- Which means they will all move into an even numbered room.
And that will leave all the odd numbered rooms.
And that's an infinite number of rooms.
And so all the new infinite number of people
can move into the odd numbered rooms.
- So then it feels like we've got twice the number of rooms,
although we're still an infinity.
In fact, the hotel can accommodate all the guests
from an infinite number of infinite hotels.
But you'll have to step in to learn how.
I guess here at Hilbert's Hotel,
there's always room for one more.
(bell rings)