Hilbert’s paradox of Grand Hotel

INTRODUCTION

 

Hilbert’s Hotel Paradox or Hilbert’s paradox of the Grand Hotel is a thought experiment proposed by German mathematician, David Hilbert  in a 1924 lecture “Über das Unendliche” reprinted and popularized through George Gamow’s 1947 book “One..Two..Three.. Infinity” . It re-explores the bizarre nature of infinity through an interesting problems of infinite guests(in several sub-cases) checking into an hotel consisting of infinite rooms.

FOR A FINITE NUMBER OF ADDITIONAL GUESTS

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Imagine a grand hotel with an infinite number of rooms stacked up in an infinite number of floors such that every room is occupied with some guest. It is very natural to think that since all the rooms are  occupied, there must be no room left for another guest waiting in the lobby. In theory, this could be attained by shifting each guest to the adjacent higher numbered room i.e. guest in room 1 is shifted to 2, 2 to 3 and so on.(guest in room n is shifted to room n+1). This can be done to as many times as required without ever running out of room.  This part of the example conveys to us in an interesting way the concept of fully-formed endlessness and how adding a finite quantity to infinity doesn’t decrease its size .

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FOR AN INFINITE NUMBER OF GUESTS IN A QUEUE

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If instead of a finite number of guests, infinite numbers of guests were to arrive in a queue, we cannot simply shift each guest by a finite number of hotel rooms, because there is always be more waiting in the queue. This can be achieved by shifting the guest in nth room to room number 2n, thereby clearing all the odd number of rooms. (cakewalk for the man in room 1, nightmare for the guest in 436648390!!)

Since, this odd number of rooms left after the shift are also infinite, it can accommodate the infinite queue. The room allotment can be understood clearly with the help of following image.

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Thus, we can see that an infinite set can be split into multiple infinite sets.

FOR AN INFINITE NUMBER OF GUESTS IN AN INFINITE NUMBER OF QUEUES

If there arrives  further a countably infinite number of guests in a countably infinite queues, how will each of them be occupied? Around 300 BC, Euclid proved that there exists an infinite number of prime numbers in the number line. If this is true, we can assign a prime number from 2 to each of those queues( and call it queue number) without ever running out of primes. We also assign a positive integer starting from two to guests in each queue. In this scenario, guest number n of queue q, will be allotted room number qn.

Considering a case, for the first queue we assign the queue number 2. Thus, the guests  in the first queue(prime number 2) will be checked in rooms 2,4,8,16 and so on. Similarly, for the second queue (prime number 3), the rooms are 3,9,27…. This way, we can ensure that no two guests will be assigned the same room number. Several rooms, that is not a power of primes will be left out, but the hotel won’t run out of rooms.

The type of infinity, as discussed in  Hilbert’s paradox is the smallest type of infinity, known as the countable infinity. A few examples of countable infinity are natural numbers, integers and whole number.

LIMITATION OF HILBERT’S HOTEL – THE REAL INFINITE BUS

Considering the above three cases ,it is difficult to imagine a scenario where the Hilbert’s hotel actually run out of rooms! This is possible, if a special queue arrives such that for every real number that exists in the number line, there is a person standing in the queue. This type of queue has uncountable infinite number of person. This type of “infinity” is larger than the ones we have discussed so far and is still infinite inside a smaller range. To understand why this type of infinity is different, consider the hotel rooms first. Each hotel room is distinct and are atomic in nature. i.e. we cannot use half a room or one-fourth a room. Although the entire set of hotel rooms are infinite, between two defined room numbers, there exists a finite number of rooms. This is not the case with the special queue, between any two guests, no matter how small the difference in their numbers, there will be an infinite number of guests inbetween!!!(Try and assign real numbers to guests and see for yourself). Cantor’s diagonal argument proves sufficiently that the set of real numbers is indeed larger.

The nature of the infinity are discussed in details in some of previous articles whose links are provided below.

To get a much more detailed insight into the concept of infinity, read the previous articles:-

Exploring Infinity Part 1

Exploring Infinity Part 2 :Cantor’s Diagonal Argument

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