The Infinite Hotel Paradox - Jeff Dekofsky

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The Infinite Hotel, a thought experiment created by German mathematician David Hilbert, is a hotel with an infinite number of rooms. Easy to comprehend, right? Wrong. What if it's completely booked but one person wants to check in? What about 40? Or an infinitely full bus of people? Jeff Dekofsky solves these heady lodging issues using Hilbert's paradox.

Lesson by Jeff Dekofsky, animation by The Moving Company Animation Studio.

In the 1920's,

the German mathematician David Hilbert

devised a famous thought experiment

to show us just how hard it is

to wrap our minds
around the concept of infinity.

Imagine a hotel with an infinite
number of rooms

and a very hardworking night manager.

One night, the Infinite Hotel
is completely full,

totally booked up
with an infinite number of guests.

A man walks into the hotel
and asks for a room.

Rather than turn him down,

the night manager decides
to make room for him.

How?

Easy, he asks the guest in room number 1

to move to room 2,

the guest in room 2 to move to room 3,

and so on.

Every guest moves from room number "n"

to room number "n+1".

Since there are an infinite
number of rooms,

there is a new room
for each existing guest.

This leaves room 1 open
for the new customer.

The process can be repeated

for any finite number of new guests.

If, say, a tour bus unloads
40 new people looking for rooms,

then every existing guest just moves

from room number "n"

to room number "n+40",

thus, opening up the first 40 rooms.

But now an infinitely large bus

with a countably infinite
number of passengers

pulls up to rent rooms.

countably infinite is the key.

Now, the infinite bus
of infinite passengers

perplexes the night manager at first,

but he realizes there's a way

to place each new person.

He asks the guest in room 1
to move to room 2.

He then asks the guest in room 2

to move to room 4,

the guest in room 3 to move to room 6,

and so on.

Each current guest moves
from room number "n"

to room number "2n" --

filling up only the infinite
even-numbered rooms.

By doing this, he has now emptied

all of the infinitely many
odd-numbered rooms,

which are then taken by the people
filing off the infinite bus.

Everyone's happy and the hotel's business
is booming more than ever.

Well, actually, it is booming
exactly the same amount as ever,

banking an infinite number
of dollars a night.

Word spreads about this incredible hotel.

People pour in from far and wide.

One night, the unthinkable happens.

The night manager looks outside

and sees an infinite line
of infinitely large buses,

each with a countably infinite
number of passengers.

What can he do?

If he cannot find rooms for them,
the hotel will lose out

on an infinite amount of money,

and he will surely lose his job.

Luckily, he remembers
that around the year 300 B.C.E.,

Euclid proved that there
is an infinite quantity

of prime numbers.

So, to accomplish this
seemingly impossible task

of finding infinite beds
for infinite buses

of infinite weary travelers,

the night manager assigns
every current guest

to the first prime number, 2,

raised to the power
of their current room number.

So, the current occupant of room number 7

goes to room number 2^7,

which is room 128.

The night manager then takes the people
on the first of the infinite buses

and assigns them to the room number

of the next prime, 3,

raised to the power of their seat
number on the bus.

So, the person in seat
number 7 on the first bus

goes to room number 3^7

or room number 2,187.

This continues for all of the first bus.

The passengers on the second bus

are assigned powers of the next prime, 5.

The following bus, powers of 7.

Each bus follows:

powers of 11, powers of 13,

powers of 17, etc.

Since each of these numbers

only has 1 and the natural number powers

of their prime number base as factors,

there are no overlapping room numbers.

All the buses' passengers
fan out into rooms

using unique room-assignment schemes

based on unique prime numbers.

In this way, the night
manager can accommodate

every passenger on every bus.

Although, there will be
many rooms that go unfilled,

like room 6,

since 6 is not a power
of any prime number.

Luckily, his bosses
weren't very good in math,

so his job is safe.

The night manager's strategies
are only possible

because while the Infinite Hotel
is certainly a logistical nightmare,

it only deals with the lowest
level of infinity,

mainly, the countable infinity
of the natural numbers,

1, 2, 3, 4, and so on.

Georg Cantor called this level
of infinity aleph-zero.

We use natural numbers
for the room numbers

as well as the seat numbers on the buses.

If we were dealing
with higher orders of infinity,

such as that of the real numbers,

these structured strategies
would no longer be possible

as we have no way
to systematically include every number.

The Real Number Infinite Hotel

has negative number rooms in the basement,

fractional rooms,

so the guy in room 1/2 always suspects

he has less room than the guy in room 1.

Square root rooms, like room radical 2,

and room pi,

where the guests expect free dessert.

What self-respecting night manager
would ever want to work there

even for an infinite salary?

But over at Hilbert's Infinite Hotel,

where there's never any vacancy

and always room for more,

the scenarios faced by the ever-diligent

and maybe too hospitable night manager

serve to remind us of just how hard it is

for our relatively finite minds

to grasp a concept as large as infinity.

Maybe you can help tackle these problems

after a good night's sleep.

But honestly, we might need you

to change rooms at 2 a.m.