Kind: captions
Language: en
even with the Advent of calculus
Infinity itself in mathematics remains
poorly understood
it was only in the late 19th century
that new mind-bending ideas helped tame
that strange Beast Infinity
[Music]
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
[Music]
the helmet 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
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 them to go to the end you have to
give them a room number
and all the rooms are full
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
okay how about the person in room one
moves into room two and the person in
room 2 moves into room 3 and the person
in room three moves into room 4 and so
on
[Music]
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
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
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
at Infinity
in fact the hotel can accommodate all
the guests from an infinite number of
infinite hotels but you'll have to stop
in to learn how
I guess here at Hilbert's Hotel there's
always room
for one more