The 7 Dimensions of Everything and Nothing ! Our 3, or more, Dimensional Universe ! Hilbert's Grand Hotel



Hilbert's Grand Hotel

The Infinite Hotel Paradox - Jeff Dekofsky


Physics and numbers go together. Numbers help us to understand the deeper unseen processes that happen in the physical world that we live in and take the mystery out of every day events. We look at the world around us and try to make sense of what we see. We have an idea, feeling or intuition about something. We observe, experiment and take measurements. Unfortunately numbers can sometimes be just as mysterious as those every day events that we are trying to unravel. Sometimes numbers just don’t balance out, cannot be resolved or point to something else that we don’t understand. Numbers in Quantum Physics can be counted but can they be measured? Quantum Physics says both yes and no.

Because numbers, no matter how big or small, are subjective beasts, they can only go so far. When they stop we need to think outside of the box to fill in the blanks. Infinity and Zero have filled in some of the blanks that have allowed us to get where we are now with technology. Everything (Infinity or ≠0) and Nothing (or 0) are used as values in most things we take for granted these days. It doesn’t matter how hard we try we can never physically get to Infinity or Zero, but, they are both essential in describing a set of something that can’t be measured until someone devises a way to measure it, for example our Universe or even Pi.

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. I remenber reading about the early European explorers that first visited Australia. They came back with stories of strange animals including the Black Swan. They were ridiculed and treated as hoaxers. Nobody believed them.

If I asked you how many green and black striped oranges or apples were in a barrel of oranges, how many would you say? If you said “0” or “None” does that mean that green and black striped oranges or 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 has’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.

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.

An Infinite set of numbers cannot be defined. We can define different subsets of types of numbers (or groups of similar values and/or properties. No two or more numbers will have exactly the same value AND property) of numbers just as we can define different subsets of people in the hotel or the buses into groups of similar ages, gender, hight etc. depending on what we are measuring or counting. But can we ever measure or count the total number of members in each subset? Quantum Physics would probally say both yes and no. It depends on what we are measuring or counting. For example 1, 2/2, 3/3 etc. If we measure 1, 2/2 and 3/3 etc. we end up with the same numbers, but if we count 1, 2/2 and 3/3 etc. we have different numbers. 

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 (The 7 Dimensions of Everything and Nothing).

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 = {} = Staff of the hotel, drivers of the infinite busses etc...
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

Hotel:
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.

Hotel:
Ground Level or Level 0 or {} = Empty Set. Rooms not 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 four 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. The value of Pi could be considered an unmeasurable set because it can’t be measured.

2.      
Uncountable Sets:
Which is the bigger set, the infinite number of people in the buses or the infinite number of people in the Hotel?
Any set of numbers, values or objects can be counted, even if they are not there. We have a bucket of oranges. 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.

3.     
Subsets:
How many numbers, values or objects can you count between 1 and 2? Would you ever get to the end?  How many numbers, values or objects can you count between 1 and 200? Would you ever get to the end? Does this mean that the set of numbers, values or objects between 1 and 200 is greater than the set of numbers, values or objects between 1 and 2, or, does this mean that both sets are subsets of the of the set of numbers, values or objects? Or, are the numbers, values or objects just a set of infinite subsets of itself.

The people in the hotel and the people in the busses can be treated as two subsets of the set of all the people and two empty sets. The hotel is a set that contains 1 subset of infinite
guest rooms and 1 Empty subset of rooms not available for guests. The infinite number of buses each contain 1 subset of infinite passenger seats and 1 Empty subset of seats not available for passengers.

4.   Empty Sets:

When the infinite number of people leave the
buses or the hotel 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?

Our Universe has often been referred to as Hilbert Space...

"The real world simply is quantum-mechanical from the start; it’s not a quantization of some classical system. The universe is described by an element of Hilbert space. All of our usual classical notions should be derived from that, not the other way around. Even space itself. We think of the space through which we move as one of the most basic and irreducible constituents of the real world, but it might be better thought of as an approximate notion that emerges at large distances and low energies." (Space Emerging from Quantum Mechanics Posted on July 18, 2016 by Sean Carroll)

Our 3, or more, Dimensional Universe
The 7 Dimensions of Everything and Nothing takes the idea of a Quantum Dimension to another level, Paradigm or Dimension. I am proposing that the idea of a Quantum Dimension is a set of conditions or properties that allow us to measure something in our reality rather than the coordinates of the object that we are measuring in a 2 or more dimensional space. Dimensions become sets of conditions or properties that pre-exist and build on each other to form our 3 (or more) dimensional reality just as length, width, and height build on each other to form something we can measure.

Our classical understanding of the properties of a set of dimensions, or coordinates, is something that can be defined and measured within our 3 (or more depending on the point of view) dimensional reality.

"The only real defining property is that they have a length, width, and height. If you place them on a Cartesian plane, you will have a X,Y, and Z axis - three axes means 3 dimensions. Aside from that, 3 dimensional objects tend to be built from 2 dimensional objects that form a “net” - a building block." (Oct 8, 2017 What are the properties of three dimensional shapes? - Quora)

But what if a set of Dimensions was a set of conditions or properties that allow us to measure something in our reality rather than the coordinates of the object that we are measuring in a 2 or more dimensional space? For example, we can measure something in relationship to something else (a ruler or scales) and come up with a measurement. But what are we measuring? How do we define it? What is that measurement related to? We need a standard or set of conditions to be able to make sense of that measurement in the first place before we do anything else. We can't measure it if we have nothing to compaire it to. What are the Demensions (conditions or properties) that allow us to measure something and make sense of the measurement? You will probally say length, width, and height. But where do they come from? What conditions or properties allow us to make the measurement in the first place? How can we measure something that is not within our 3 or more dimensional reality with the tools that are a part of our 3 or more dimensional reality? Its a bit like trying to take a photograph with a hammer.

"The properties of a quantum dot are determined by size, shape, composition, and structure. The interesting electronic properties of quantum dots arise from the specific size of their energy band gaps. Dec 19, 2018 - The Properties and Applications of Quantum Dots" (Dec 19, 2018 - The Properties and Applications of Quantum Dots)

It seems to me that the idea of a Dimension takes on a whole new Paradigm (or Perspective). Dimensions become sets of conditions or properties that pre-exist and build on each other to form our 3 (or more) dimensional reality just as length, width, and height build on each other to form something we can measure.



Peter S Anderson
Perth, Western Australia



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