schöne hotel badezimmer

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 mindsaround the concept of infinity. imagine a hotel with an infinitenumber of rooms and a very hardworking night manager. one night, the infinite hotelis completely full, totally booked upwith an infinite number of guests.

a man walks into the hoteland asks for a room. rather than turn him down, the night manager decidesto 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 infinitenumber of rooms, there is a new roomfor each existing guest. this leaves room 1 openfor the new customer. the process can be repeated for any finite number of new guests. if, say, a tour bus unloads40 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 infinitenumber of passengers pulls up to rent rooms. countably infinite is the key. now, the infinite busof 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 1to 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, each current guest movesfrom room number "n" to room number "2n" -- filling up only the infiniteeven-numbered rooms. by doing this, he has now emptied

all of the infinitely manyodd-numbered rooms, which are then taken by the peoplefiling off the infinite bus. everyone's happy and the hotel's businessis booming more than ever. well, actually, it is boomingexactly the same amount as ever, banking an infinite numberof 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 lineof infinitely large buses, each with a countably infinitenumber 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 remembersthat around the year 300 b.c.e., euclid proved that thereis an infinite quantity of prime numbers.

so, to accomplish thisseemingly impossible task of finding infinite bedsfor infinite buses of infinite weary travelers, the night manager assignsevery current guest to the first prime number, 2, raised to the powerof 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 peopleon the first of the infinite buses and assigns them to the room number of the next prime, 3, raised to the power of their seatnumber on the bus. so, the person in seatnumber 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' passengersfan out into rooms using unique room-assignment schemes based on unique prime numbers. in this way, the nightmanager can accommodate every passenger on every bus. although, there will bemany rooms that go unfilled, like room 6, since 6 is not a powerof any prime number. luckily, his bossesweren't very good in math,

so his job is safe. the night manager's strategiesare only possible because while the infinite hotelis certainly a logistical nightmare, it only deals with the lowestlevel of infinity, mainly, the countable infinityof the natural numbers, 1, 2, 3, 4, and so on. georg cantor called this levelof infinity aleph-zero. we use natural numbersfor the room numbers as well as the seat numbers on the buses.

if we were dealingwith higher orders of infinity, such as that of the real numbers, these structured strategieswould no longer be possible as we have no wayto 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 managerwould 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.