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• 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 countedly infinite number of passengers

• pulls up to rent rooms.

• Countedly 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 one.

• 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.

• 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 countedly 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 accomodate

• 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.

In the 1920's,

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B2 H-INT TED-Ed infinite room bus hotel guest

# 【TED-Ed】The Infinite Hotel Paradox - Jeff Dekofsky

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Sofi posted on 2014/04/09
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