WEBVTT

00:00:06.531 --> 00:00:07.691
In the 1920's,

00:00:07.715 --> 00:00:10.184
the German mathematician David Hilbert

00:00:10.208 --> 00:00:12.437
devised a famous thought experiment

00:00:12.461 --> 00:00:14.191
to show us just how hard it is

00:00:14.215 --> 00:00:17.665
to wrap our minds
around the concept of infinity.

00:00:18.353 --> 00:00:21.659
Imagine a hotel with an infinite
number of rooms

00:00:21.683 --> 00:00:23.988
and a very hardworking night manager.

00:00:24.528 --> 00:00:27.523
One night, the Infinite Hotel
is completely full,

00:00:27.547 --> 00:00:31.003
totally booked up
with an infinite number of guests.

00:00:31.027 --> 00:00:34.161
A man walks into the hotel
and asks for a room.

00:00:34.185 --> 00:00:35.444
Rather than turn him down,

00:00:35.468 --> 00:00:37.907
the night manager decides
to make room for him.

00:00:37.931 --> 00:00:38.947
How?

00:00:38.971 --> 00:00:41.635
Easy, he asks the guest in room number 1

00:00:41.659 --> 00:00:43.711
to move to room 2,

00:00:43.735 --> 00:00:46.056
the guest in room 2 to move to room 3,

00:00:46.080 --> 00:00:47.138
and so on.

00:00:47.449 --> 00:00:49.838
Every guest moves from room number "n"

00:00:49.862 --> 00:00:52.179
to room number "n+1".

00:00:52.721 --> 00:00:54.812
Since there are an infinite
number of rooms,

00:00:54.836 --> 00:00:57.009
there is a new room
for each existing guest.

00:00:57.413 --> 00:00:59.760
This leaves room 1 open
for the new customer.

00:00:59.784 --> 00:01:01.070
The process can be repeated

00:01:01.094 --> 00:01:03.511
for any finite number of new guests.

00:01:03.535 --> 00:01:07.529
If, say, a tour bus unloads
40 new people looking for rooms,

00:01:07.553 --> 00:01:09.642
then every existing guest just moves

00:01:09.666 --> 00:01:10.980
from room number "n"

00:01:11.004 --> 00:01:13.638
to room number "n+40",

00:01:13.662 --> 00:01:16.200
thus, opening up the first 40 rooms.

00:01:17.157 --> 00:01:19.171
But now an infinitely large bus

00:01:19.195 --> 00:01:21.744
with a countably infinite
number of passengers

00:01:21.768 --> 00:01:23.673
pulls up to rent rooms.

00:01:23.697 --> 00:01:25.686
countably infinite is the key.

00:01:26.164 --> 00:01:28.530
Now, the infinite bus
of infinite passengers

00:01:28.554 --> 00:01:30.518
perplexes the night manager at first,

00:01:30.542 --> 00:01:32.010
but he realizes there's a way

00:01:32.034 --> 00:01:33.349
to place each new person.

00:01:33.373 --> 00:01:36.391
He asks the guest in room 1
to move to room 2.

00:01:36.415 --> 00:01:38.527
He then asks the guest in room 2

00:01:38.551 --> 00:01:40.435
to move to room 4,

00:01:40.459 --> 00:01:42.809
the guest in room 3 to move to room 6,

00:01:42.833 --> 00:01:44.105
and so on.

00:01:44.129 --> 00:01:47.313
Each current guest moves
from room number "n"

00:01:47.337 --> 00:01:49.029
to room number "2n" --

00:01:50.807 --> 00:01:54.060
filling up only the infinite
even-numbered rooms.

00:01:54.084 --> 00:01:55.929
By doing this, he has now emptied

00:01:55.953 --> 00:01:58.867
all of the infinitely many
odd-numbered rooms,

00:01:58.891 --> 00:02:02.506
which are then taken by the people
filing off the infinite bus.

00:02:03.242 --> 00:02:06.875
Everyone's happy and the hotel's business
is booming more than ever.

00:02:06.899 --> 00:02:10.416
Well, actually, it is booming
exactly the same amount as ever,

00:02:10.440 --> 00:02:12.923
banking an infinite number
of dollars a night.

00:02:14.076 --> 00:02:16.355
Word spreads about this incredible hotel.

00:02:16.379 --> 00:02:18.544
People pour in from far and wide.

00:02:18.568 --> 00:02:20.842
One night, the unthinkable happens.

00:02:20.866 --> 00:02:23.407
The night manager looks outside

00:02:23.431 --> 00:02:27.517
and sees an infinite line
of infinitely large buses,

00:02:27.541 --> 00:02:30.329
each with a countably infinite
number of passengers.

00:02:30.353 --> 00:02:31.386
What can he do?

00:02:31.410 --> 00:02:34.207
If he cannot find rooms for them,
the hotel will lose out

00:02:34.231 --> 00:02:35.958
on an infinite amount of money,

00:02:35.982 --> 00:02:37.955
and he will surely lose his job.

00:02:37.979 --> 00:02:41.790
Luckily, he remembers
that around the year 300 B.C.E.,

00:02:41.814 --> 00:02:44.726
Euclid proved that there
is an infinite quantity

00:02:44.750 --> 00:02:46.634
of prime numbers.

00:02:47.372 --> 00:02:49.660
So, to accomplish this
seemingly impossible task

00:02:49.684 --> 00:02:52.285
of finding infinite beds
for infinite buses

00:02:52.309 --> 00:02:54.291
of infinite weary travelers,

00:02:54.315 --> 00:02:57.182
the night manager assigns
every current guest

00:02:57.206 --> 00:02:59.042
to the first prime number, 2,

00:02:59.066 --> 00:03:01.867
raised to the power
of their current room number.

00:03:01.891 --> 00:03:04.535
So, the current occupant of room number 7

00:03:04.559 --> 00:03:07.541
goes to room number 2^7,

00:03:07.565 --> 00:03:09.261
which is room 128.

00:03:10.236 --> 00:03:13.757
The night manager then takes the people
on the first of the infinite buses

00:03:13.781 --> 00:03:15.806
and assigns them to the room number

00:03:15.830 --> 00:03:18.291
of the next prime, 3,

00:03:18.315 --> 00:03:21.728
raised to the power of their seat
number on the bus.

00:03:21.752 --> 00:03:25.259
So, the person in seat
number 7 on the first bus

00:03:25.283 --> 00:03:28.360
goes to room number 3^7

00:03:28.384 --> 00:03:31.610
or room number 2,187.

00:03:31.634 --> 00:03:34.069
This continues for all of the first bus.

00:03:34.093 --> 00:03:35.741
The passengers on the second bus

00:03:35.765 --> 00:03:39.410
are assigned powers of the next prime, 5.

00:03:39.434 --> 00:03:41.493
The following bus, powers of 7.

00:03:41.517 --> 00:03:42.921
Each bus follows:

00:03:42.945 --> 00:03:44.746
powers of 11, powers of 13,

00:03:44.770 --> 00:03:46.799
powers of 17, etc.

00:03:47.370 --> 00:03:48.729
Since each of these numbers

00:03:48.753 --> 00:03:50.968
only has 1 and the natural number powers

00:03:50.992 --> 00:03:53.213
of their prime number base as factors,

00:03:53.237 --> 00:03:55.386
there are no overlapping room numbers.

00:03:55.410 --> 00:03:58.339
All the buses' passengers
fan out into rooms

00:03:58.363 --> 00:04:00.846
using unique room-assignment schemes

00:04:00.870 --> 00:04:03.486
based on unique prime numbers.

00:04:03.510 --> 00:04:05.701
In this way, the night
manager can accommodate

00:04:05.725 --> 00:04:07.846
every passenger on every bus.

00:04:07.870 --> 00:04:11.064
Although, there will be
many rooms that go unfilled,

00:04:11.088 --> 00:04:12.356
like room 6,

00:04:12.380 --> 00:04:15.095
since 6 is not a power
of any prime number.

00:04:15.119 --> 00:04:17.512
Luckily, his bosses
weren't very good in math,

00:04:17.536 --> 00:04:18.875
so his job is safe.

00:04:19.507 --> 00:04:22.007
The night manager's strategies
are only possible

00:04:22.031 --> 00:04:26.624
because while the Infinite Hotel
is certainly a logistical nightmare,

00:04:26.648 --> 00:04:29.957
it only deals with the lowest
level of infinity,

00:04:29.981 --> 00:04:33.513
mainly, the countable infinity
of the natural numbers,

00:04:33.537 --> 00:04:36.594
1, 2, 3, 4, and so on.

00:04:36.618 --> 00:04:40.513
Georg Cantor called this level
of infinity aleph-zero.

00:04:40.945 --> 00:04:43.042
We use natural numbers
for the room numbers

00:04:43.066 --> 00:04:45.188
as well as the seat numbers on the buses.

00:04:45.913 --> 00:04:48.252
If we were dealing
with higher orders of infinity,

00:04:48.276 --> 00:04:49.848
such as that of the real numbers,

00:04:49.872 --> 00:04:52.844
these structured strategies
would no longer be possible

00:04:52.868 --> 00:04:56.405
as we have no way
to systematically include every number.

00:04:57.002 --> 00:04:58.803
The Real Number Infinite Hotel

00:04:58.827 --> 00:05:00.905
has negative number rooms in the basement,

00:05:00.929 --> 00:05:02.364
fractional rooms,

00:05:02.388 --> 00:05:04.484
so the guy in room 1/2 always suspects

00:05:04.508 --> 00:05:07.181
he has less room than the guy in room 1.

00:05:07.205 --> 00:05:10.308
Square root rooms, like room radical 2,

00:05:10.332 --> 00:05:11.438
and room pi,

00:05:11.462 --> 00:05:14.325
where the guests expect free dessert.

00:05:14.349 --> 00:05:17.374
What self-respecting night manager
would ever want to work there

00:05:17.398 --> 00:05:19.005
even for an infinite salary?

00:05:19.029 --> 00:05:20.918
But over at Hilbert's Infinite Hotel,

00:05:20.942 --> 00:05:22.420
where there's never any vacancy

00:05:22.444 --> 00:05:24.004
and always room for more,

00:05:24.028 --> 00:05:26.926
the scenarios faced by the ever-diligent

00:05:26.950 --> 00:05:28.760
and maybe too hospitable night manager

00:05:28.784 --> 00:05:31.490
serve to remind us of just how hard it is

00:05:31.514 --> 00:05:33.905
for our relatively finite minds

00:05:33.929 --> 00:05:36.767
to grasp a concept as large as infinity.

00:05:37.132 --> 00:05:39.083
Maybe you can help tackle these problems

00:05:39.107 --> 00:05:40.403
after a good night's sleep.

00:05:40.427 --> 00:05:42.276
But honestly, we might need you

00:05:42.300 --> 00:05:44.701
to change rooms at 2 a.m.