HOTEL INFINITY : An Infinite Hotel, that is, a hotel with infinitely

Compiled by

VIVEK KUMAR

Imagine that you arrive at an infinite hotel, that is, a hotel with infinitely many rooms in it.

You need a room, the only problem for the receptionist (besides perhaps having a big guest book) is that the hotel is fully occupied and now he needs to find a room for you.
But the receptionist is clever. He tells the guest in room 1 to move to room 2 and the one in room 2 to move to room 3 and in general, the guest in room n moves to room n+1.
We can imagine they all move rooms at the same time.
When that’s done, there is a room available at room 1 and all guests are accommodated.
Great! What this thought experiment shows is that ∞ + 1 = ∞.
In a fully booked hotel with infinitely many rooms, you can always find a room for one more.
The Genius of David Hilbert
David Hilbert – one of the greatest and most prolific mathematicians of the 20th century, invented this analogy to explain the contra-intuitiveness of infinite sets and transfinite arithmetic in a lecture “Über das Unendliche” in 1924.
Hilbert wanted to show how to think of infinity and how to work with it in the right way. His message was “not to be afraid of it”, but rather to welcome its many facets in a structured way.
This had historically been a mathematical taboo and the first mathematicians that tried to set infinity on solid ground such as Georg Cantor were heavily criticized by their contemporaries. Hilbert however saw the beauty in this theory!
Some of the issues come from the contra-intuitiveness of infinite sets. As Wikipedia puts it:
The statements “there is a guest to every room” and “no more guests can be accommodated” are not equivalent when there are infinitely many rooms.
We need to note that one should be a little careful with the statement ∞ + 1 = ∞ because infinity is not a real or complex number and therefore the above identity doesn’t make any sense unless we have defined what “number space” we are working in.
However, we assume of course that we are in an extended space like the extended real line of the extended complex plane where we are allowed to add and subtract numbers to and from ∞.

Back to the thought experiment. What if k number of people arrives at the fully booked infinite hotel and seeks k rooms?
No problem, we just tell each guest to move to the room having their number plus k. Then the first k rooms will be available, and all the guests will have rooms.
For example, if 3 people were to visit the hotel and wanted a room each, then the person in room 1 would be told to move to room 4, the one in room 2 to room 5 etc.
Now we’re at it, how about the question in the subheading of this article? How would we make sense of ∞ – 1? Well, in the infinite hotel we can simply ask the person in room number 1 to leave the hotel and then move the person in room 2 to room 1, the person in room 3 to room 2 and so forth yielding ∞ yet again.
That is, ∞ – 1 = ∞.
Into Infinity
This is all fine but what if an infinite number of new guests arrive at the hotel? When we say infinite here, we really mean countably infinite which is a mathy way of saying that there is a way to label each one by natural (positive and whole) numbers.
This time, we can’t just shift the guests as we did before because that would require an infinite shift. However, we can solve this by moving the guest in room 1 to room 2, the guest in room 2 to room 4 and in general by moving the guest in room n to room 2n.
We don’t care about the fact that some of the guests in big room numbers need to move to a room which is probably quite far away, we only care about the underlying mathematics of course.
Now all the odd-numbered rooms are available and we can place the infinitely many new guests in those rooms.
What this is really saying is that the “size” (called the cardinality) of the infinite set of natural numbers is the same as the cardinality of the even natural numbers (despite being a subset of the natural numbers of course).
Wait… Does that mean that there are just as many natural numbers as there are even natural numbers?
Well… In fact, yes…
In mathematics, we compare infinite sets by considering functions between them and in this situation we can define a function f(n) = 2n from the natural numbers to the even natural numbers.

Think about it, if you don’t know how to count but need to compare two piles of stones, how would you do it? Your only job is to determine which pile contains the greatest number of stones.
Well, you could pair them up by repeatedly taking one from each pile and until there is only one or zero piles left. At that point, the process terminates.
If there is a single pile left, then that was the one with the greatest number of stones originally and if they match up completely yielding no piles when the process terminates, then that means that all their stones could be paired up two by two and they would have had the same number of stones in them to start with.
When we generalize this concept to infinite sets, we get exactly the scenario from above with the stones. The pairing is just done with bijective functions instead of small stones.
Some Infinities are Bigger than Others
What if an infinite number of busses each containing an infinite number of persons arrive at the fully booked hotel and they all want a room?
No problem!!
We can label each bus and each seat on each bus by natural numbers (after all, we are dealing with countably infinite sets). Then each person has a unique “address” in the form of two numbers: one number s which is the seat number on the bus and one number b which is the bus number. The persons already in the hotel have b=0.
Then simply put each person in room 2^s ⋅ 3^b.
For example, the person in room 1119744 = 2⁹⋅ 3⁷ was sitting in bus number 7 and seat number 9. This can easily be generalized to many more layers of infinity using more primes.
For instance, if an infinite number of ferries each carrying an infinite number of busses each having an infinite number of passengers, we can fit them into our hotel by extension of the above. We simply add the prime number 5 into the algorithm mutatis mutandis.
Now we have three layers of infinite nesting and any finite layers can be solved in this way — after all, there are infinitely many prime numbers, so we won’t exactly run out of them.
The prime factorization method is just one of several other methods of solving this problem. The important thing is of course that it can be solved. But how about an infinite number of nested layers of infinity?
It turns out that this problem cannot always be solved!
That’s because some infinities are bigger than others!
Yes, you read that right. Infinity is not just infinity when it comes to the “size” or cardinality of sets.
For example, the cardinality of the set of all fractions of whole numbers is the same as the cardinality of the natural numbers.
Yes, there are as many fractions of whole numbers as there are whole positive numbers although the positive whole numbers are fractions themselves…!
We can prove this by finding a bijection between the two sets. That is, to find a one-to-one map between the sets pairing up each element from one of the sets to an element in the other. I will leave this as a small exercise to the reader.
If there exists a bijection between the natural numbers (positive and whole numbers) and a given set A, then A is said to be countable and the cardinality is that of the natural numbers denoted ℕ. This cardinality is sometimes denoted ℵ0.
The Continuum
Okay, so some infinities are greater than others but that leaves a trail of interesting questions behind right?
What are examples of sets with greater cardinality than the natural numbers? That is, what does an uncountable set “look” like? And how many different infinities are there? are just some of the questions that come to mind.
A good example of a set with a greater cardinality than that of the natural numbers is the set of real numbers denoted ℝ. This set is uncountable, meaning that there does not exist a bijection between ℝ and ℕ.
The set of real numbers includes, of course, all the fractions but it also contains numbers like π and e which cannot be written as fractions of whole numbers.
The question of whether there is an infinity greater than the cardinality of the natural numbers but less than that of the real numbers is called the continuum hypothesis and it has been shown that it can be proved neither true nor false within our axiomatic system.
This then becomes borderline philosophy and we are entering the field of metamathematics and mathematical logic where we can discuss if we should be using alternate axiomatic systems in order to solve this problem. This will in turn create an alien form of mathematics. Quite intriguing if you ask me!
Oh, and there are in fact infinitely many different kinds of infinities; the question now is, of course, which kind of infinity??