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In the 1920's,

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the German mathematician David Hilbert

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devised a famous thought experiment

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to show us just how hard it is

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to wrap our minds
around the concept of infinity.

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Imagine a hotel with an infinite
number of rooms

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and a very hardworking night manager.

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One night, the Infinite Hotel
is completely full,

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totally booked up
with an infinite number of guests.

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

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Rather than turn him down,

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the night manager decides
to make room for him.

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How?

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Easy, he asks the guest in room number 1

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to move to room 2,

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the guest in room 2 to move to room 3,

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and so on.

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Every guest moves from room number "n"

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to room number "n+1".

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Since there are an infinite
number of rooms,

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there is a new room
for each existing guest.

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This leaves room 1 open
for the new customer.

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The process can be repeated

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for any finite number of new guests.

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

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then every existing guest just moves

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from room number "n"

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to room number "n+40",

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thus, opening up the first 40 rooms.

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But now an infinitely large bus

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with a countably infinite
number of passengers

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pulls up to rent rooms.

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countably infinite is the key.

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Now, the infinite bus
of infinite passengers

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perplexes the night manager at first,

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but he realizes there's a way

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to place each new person.

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

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He then asks the guest in room 2

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to move to room 4,

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the guest in room 3 to move to room 6,

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and so on.

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Each current guest moves
from room number "n"

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to room number "2n" --

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filling up only the infinite
even-numbered rooms.

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By doing this, he has now emptied

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all of the infinitely many
odd-numbered rooms,

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

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

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

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banking an infinite number
of dollars a night.

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Word spreads about this incredible hotel.

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People pour in from far and wide.

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One night, the unthinkable happens.

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The night manager looks outside

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and sees an infinite line
of infinitely large buses,

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each with a countably infinite
number of passengers.

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What can he do?

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

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on an infinite amount of money,

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and he will surely lose his job.

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

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Euclid proved that there
is an infinite quantity

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of prime numbers.

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So, to accomplish this
seemingly impossible task

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of finding infinite beds
for infinite buses

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of infinite weary travelers,

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the night manager assigns
every current guest

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to the first prime number, 2,

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raised to the power
of their current room number.

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So, the current occupant of room number 7

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goes to room number 2^7,

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which is room 128.

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

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and assigns them to the room number

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of the next prime, 3,

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

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

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goes to room number 3^7

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or room number 2,187.

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This continues for all of the first bus.

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The passengers on the second bus

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are assigned powers of the next prime, 5.

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The following bus, powers of 7.

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Each bus follows:

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powers of 11, powers of 13,

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powers of 17, etc.

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Since each of these numbers

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only has 1 and the natural number powers

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of their prime number base as factors,

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there are no overlapping room numbers.

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All the buses' passengers
fan out into rooms

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using unique room-assignment schemes

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based on unique prime numbers.

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In this way, the night
manager can accommodate

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every passenger on every bus.

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Although, there will be
many rooms that go unfilled,

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like room 6,

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since 6 is not a power
of any prime number.

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

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so his job is safe.

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The night manager's strategies
are only possible

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because while the Infinite Hotel
is certainly a logistical nightmare,

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it only deals with the lowest
level of infinity,

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mainly, the countable infinity
of the natural numbers,

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

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Georg Cantor called this level
of infinity aleph-zero.

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We use natural numbers
for the room numbers

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as well as the seat numbers on the buses.

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If we were dealing
with higher orders of infinity,

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such as that of the real numbers,

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these structured strategies
would no longer be possible

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as we have no way
to systematically include every number.

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The Real Number Infinite Hotel

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has negative number rooms in the basement,

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fractional rooms,

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so the guy in room 1/2 always suspects

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he has less room than the guy in room 1.

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Square root rooms, like room radical 2,

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and room pi,

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where the guests expect free dessert.

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

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even for an infinite salary?

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But over at Hilbert's Infinite Hotel,

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where there's never any vacancy

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and always room for more,

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the scenarios faced by the ever-diligent

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and maybe too hospitable night manager

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serve to remind us of just how hard it is

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for our relatively finite minds

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to grasp a concept as large as infinity.

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Maybe you can help tackle these problems

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after a good night's sleep.

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But honestly, we might need you

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to change rooms at 2 a.m.