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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.
