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 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 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 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 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. 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 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 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 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.

There is no paradox here.

How can a hotel with an infinite number of rooms be full?

If the infinite hotel is full, wouldn't that mean that there couldn't be any more new guests? Since infinity has been reached and all people are in the hotel?

If the hotel has infinite rooms and pushing each guest at room n to room n+1 makes room 1 open and the manager only deals with 1 person at a time cant he just move everybody 1 room over every time a new person comes in, or he could just pick a very large number and move everyone that many rooms over.

You wouldn't have to move anyone out of a room, since there would be an infinite number of rooms, the hotel could accomidate an infinite number of guests. Infinity is not an actual reachable figure, mearly a representation of the possibility of endlessness.

I like maths.

There is no such thing as infinite. -Jayden Smith