Was Hotel California written keeping Hilbert's Hotel in mind? A peek into the Infinite Hotel Paradox

Recently we were listening to "Hotel California", the Eagles classic, featuring one of the greatest guitarists of all time, Don Felder. The song is great, but lie the gees we are, something in the lyrics caught our ear:

"Welcome to the Hotel California

  Such a lovely place (such a lovely place)

  Such a lovely face.

  Plenty of room at the Hotel California

  Any time of year (any time of year) you can find it here"


The line in bold triggered something in our memory, something we read about when we started learning the concepts of sets and different types of infinities: The Hilbert Infinite Hotel Paradox.
We wonder whether the songwriters deliberately wrote this. Anyway, this led us to revisit this paradoxical problem. 

The Problem:

Consider a hypothetical hotel with countably infinite number of rooms. A lone traveler seeing shelter arrives at the hotel, only to find all rooms occupied. Yet, he is given a room to stay for the night by the manager. How is that possible?

Solution:

The solution is based on the concept of a countably infinite set. A set is said to be countably infinite if it has a one-to-one correspondence with the natural numbers. One might say that the new guest may be accommodated in the "last" unoccupied room. However there is no concept of a "last" room in this case, as there is always a "next" room after any given room and even if we could locate such a room, it would be occupied. We exploit this idea to propose a solution. The guest staying in room 1 is shifted to room 2, the one in room 2 is shifted to room 3 , and so on and so forth. Thus the room numbered 1 is made available for the new guest. Thus we are actually defining a bijective mapping from the set of natural numbers to the set of natural numbers excluding 1 given by

This argument can be extended to accommodate a group of k (finite positive integer) travelers arriving at the hotel. In that case the mapping is 


An (Infinitely) Bigger Problem!!

Suppose a (countably) infinitely long bus of guests arrives arrives at the hotel seeking accommodation. How is the manager supposed to give them shelter for the night?

Solution:

Yet again mathematics comes to the rescue. We shift every existing guest from their current room to a room bearing a number that is twice the number of their current room. Thus the guest staying in room 1 is shifted to room 2, the one staying in room 2 is shifted to room 4, and so on and so forth. The the existing guests occupy the even numbered rooms and the new guests occupy the odd numbered rooms. The bijection used here is nothing but 



An Even Infinitely Bigger Problem!!!

Suppose a countably infinite number of buses arrive at the hotel, each carrying a countably infinite number of guests. The manager is utterly perplexed. How can he accommodate all of his guests?

Solution:

Here we make use of the fact that there are a countably infinite number of prime numbers. We shift every current guest staying in room number to the room numbered . Let the buses be numbered 1,2,...,c,....i.e. there is a bijection between the buses and . Further let every guest on each bus be numbered as 1,2,...,s,....according to their seat numbers i.e. every guest belong to a unique bus and has a unique seat number assigned to him/her. Now each guest belonging to the c-th bus and having the s-th seat number in that bus is assigned the room numbered . Since each room number has a unique prime factorization, each guest is assigned a different room. This mapping is different from the previous mapping in the sense that many rooms between two occupied rooms remain empty.


Thought this is the end? Think again!

 If there are a finite(say k) number of nested modes of transportation then this idea can be extended to provide accommodation to all the guests. To fix ideas, say for example, the hotel is situated beside a river and a a countably infinite number of ferries each carrying countably infinite number of buses and countably infinite number of guests in each bus. the room number  is assigned to the s-th guest of the c-th bus on the f-th ferry. The general bijection can be constructed similarly using prime factorization.


References:

  1. https://medium.com/i-math/hilberts-infinite-hotel-paradox-ca388533f05
  2. Kragh ,Helge."The True(?) Story of Hilbert's Infinite Hotel". https://arxiv.org/ftp/arxiv/papers/1403/1403.0059.pdf 

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