Hilbert's Grand Hotel

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.

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.

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,

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.

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.

Subset of C = C1 = People in the hotel.

Subset of C = C1 = People in the bus.

Room
{} = Empty Set. Manager’s room etc...

Room 1 = 1st Guest,

Room 2 = 2nd Guest,

Room 3 = 3rd Guest,

Room 4 = 4th Guest...

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 {} = 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...

...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...

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.