The 7 Dimensions of Everything and Nothing

Hilbert's Grand Hotel

The Infinite Hotel Paradox - Jeff Dekofsky

Hilbert's Grand Hotel (a thought experiment, Paradigm or Paradox) describes the idea of infinite sets of infinite numbers within each other, within Infinity. There is a hotel with an infinite number of rooms with a person in each room. The hotel also has room for an infinite number of people in an infinite number of buses.

The idea of Everything only exists because something exists that can’t be measured or counted. If something can be measured or counted then it cannot be Everything. The idea of nothing exists because there is no other way to define its existence or lack of existence. If I asked you how many apples were in a barrel of oranges, how many would you say? If you said “0” or “None” does that mean that apples don’t exist?

A Set is a container (Barrel) that contains something or groups of something (Oranges or Apples) or even nothing (Empty). Just because an Empty Set does not contain what we are looking for, it does not mean that it is empty. It only means that it hasn’t been filled with what we are looking for. Maybe there is other stuff in there that we are not looking for or can’t see.

Although the experiment does not explicitly say how the people originally got into the hotel or the buses in the first place, I feel it is safe to infer that they were waiting outside the hotel/buses and were then randomly given or issued a room key/seat ticket as they entered the hotel/buses (bijection, bijective function, or one-to-one correspondence).

 First thing is to establish what we are trying to achieve; to fit all the people in each bus with the people already in the hotel into their own room.

These people could also be divided into any other number of Subsets.What Sets are we working with?

Set A = hotel,
Subset of A = {} = Rooms not available as Guest rooms, manager’s bedroom etc...
Subset of A = A1 = Guest rooms.
Subset of A = A2 = Person in each room in the hotel,

Set B = Bus
Subset of B = {} = Seats not available for passengers, driver’s seat etc...
Subset of B = B1 = Passenger Seats on the bus.
Subset of B = B2 = Person on each Seat in the bus.
Set C = Number of people
Subset of C = {} = Empty Set.
Subset of C = C1 = People in the hotel.
Subset of C = C1 = People in the bus.

If the manager took the infinite people out of the hotel at the same time as the infinite number of people got out of each bus as it arrived, he would have only one set of infinite people? He could then fit the infinite number of people outside the hotel comfortably into the infinite hotel. Each person would have a key to their own room using the above bijective function

Room {} = Empty Set. Manager’s room etc...
Room 1 = 1st Guest,
Room 2 = 2nd Guest,
Room 3 = 3rd Guest,
Room 4 = 4th Guest...

If the hotel had an infinite number of levels the manager could also fit all the infinite people that came out of each level and bus and fit them all comfortably into each corresponding level of the hotel.

Ground Level or Level {} = Empty Set. No rooms available for guests.
1st bus = Level 1, Room 1.1, 1.2, 1.3...
2nd bus = Level 2, Room 2.1, 2.2, 2.3...

This would mean that there are no empty rooms, and each person from the hotel is only disturbed once.

There are at least 5 other ways to fit the people into the hotel, however it seems to me that there will be at least 1 person always looking for a room when we use any of the these methods because every room is already occupied. There is also the possibility of ending up with rooms that are empty.

This thought experiment is actually describing three different types of infinite sets...

1.      Unmeasurable Sets: The size of the hotel/buses are unmeasurable because their dimensions/ volume cannot be measured. If the hotel/buses could be measured they would be finite. There is no such thing as half a room in the hotel or half a seat on each bus. There can only be a room or a seat, no matter how infinitely big or infinitely small. The hotel manager could easily fit all the people from one bus into one room in the hotel. Our Universe is the probably the best example of an infinite space because it can’t be measured and it is expanding faster than expected. Quantum particles are also an examples of an unmeasurable set. We can only approximate what we see.

2.      Uncountable Sets: Any set of numbers, values or objects can be counted, even if they are not there. We start from 0 for nothing and then 1 and then 2 etc. and we end up with a finite number. But, there are things that we can count and never get to the end. We can count numbers by adding 1 to the previous number and never get to the end. Could we ever count the infinite number of people in the hotel or the infinite number of buses? Will we ever get to the end? Mathematics has never been able to count the total decimal places of Pi and probably never will. The value of Pi could also be considered an unmeasurable set because it can’t be measured.

3.      Empty Sets: The hotel is a set that contains 2 sets of infinite rooms and infinite people and 1 Empty set. The infinite number of buses each contain 2 sets of infinite seats and infinite people and 1 Empty set. When the infinite number of people leave the hotel or the buses they become empty, but they still exist.

The paradox of Hilbert's Grand Hotel is no longer a paradox, but simply a way to describe something that can’t be measured and/or counted.

Does this mean that both everything and nothing fit together into one or more infinities because both the hotel and the buses are empty until they get filled? Is it possible to have something with an infinite value of nothing (empty set theory)?

The point I am trying to make is that mathematics can sometimes be like statistics where we can manipulate numbers within a mathematical model or algorithm, and according to the rules, to find the answers we are looking for. Sometimes we have to make up or discover the rules as we go along. Isn't this how mathematics and science came about in the first place? Isn't this why there are so many different theories about mathematics, statistics, Classic and sub-atomic/quantum physics?