what is Infinity paradox ?

In the 1920s 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 infinite number of guests a man walks into the hotel and asked 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 guests 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 guest of the 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 hotels 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 BCE Euclid proved 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 guests 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 to the 7th power, 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 to the 7th power or room number 2187 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 seven each bus follows powers. 11 powers of 13 powers of 17 etc, since each of these numbers only has one 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 managers strategies are only possible because while the infinite hotel is certainly a logistical nightmare, it only deals with the lowest level of infinity namely the countable Infinity of the natural numbers 1 2 3 4 and so on, Georg Cantor called this level of infinity LF 0 we use natural numbers for the room numbers as well 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 a negative number rooms in the basement fractional rooms. So the guy in room one half 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 to hospitable night manager serve to remind us of just how hard it is for are 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.

Sources https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel


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