INFINITIES – 4. No Vacancies But Rooms Still Available At Hotel Infinity

Illustration copyright MURRAY YOUNG

This is the fourth in a series of posts exploring infinity. There will be one infinity larger than another infinity, infinite perimeters, infinite regress, infinity paradoxes, an infinite set of bizarre things called cyclic numbers, and other infinity-related concepts.

A lot has been written about Hotel Infinity. Hopefully you will learn something new here. Here’s the set-up. Imagine a hotel containing fifty rooms, and there is a single guest in each room. If a new guest comes along and wishes a room she will be out of luck. Now imagine a hotel with an infinite number of rooms, with a guest in each room. Lo and behold a new guest can still be accommodated while still giving every guest their own room. It was German logician David Hilbert who came up with the idea of Hotel Infinity as a way of demonstrating the surprising and counter-intuitive properties of infinite sets (one of the people Hilbert worked with at Göttingen University in 1915 was a young man you may have heard of named Albert Einstein).


DAVID HILBERT
By Unknown author, derivative work Lämpel – Reid, Constance (1970) Hilbert, Berlin, Heidelberg: Springer Berlin Heidelberg Imprint Springer, p. 230 ISBN: 978-3-662-27132-2., Public Domain, https://commons.wikimedia.org/w/index.php?curid=130481740

Infinities can be quite perplexing. For example, the part may be the same size as the whole! This is demonstrated by the fact that there are just as many natural numbers(1, 2, 3 . . . ) as there are even numbers (2, 4, 6 . . .) because you can match 1 up with 2, 2 up with 4, 3 up with 6 and so on. The idea of Hotel Infinity was first introduced by Hilbert in his 1924 lecture “Über das Unendliche” (“On The Infinite”), and Hilbert’s idea was first popularized by George Gamow in his excellent and quite accessible book ‘One, Two, Three . . . Infinity’ published in 1947 (page 17). The celebrated Polish science fiction writer Stanislaw Lem (1921 – 2006) also wrote a story called ‘The Extraordinary Hotel, or the Thousand and First Journey of Ion the Quiet’. It was first published in 1968 as part of the anthology ‘Stories About Sets’ edited by N. Ya. Vilenkin. It takes place at the Hotel Cosmos which has an infinite number of rooms.


STANISLAW LEM IN 1966
By Courtesy of Lem’s secretary, Wojciech Zemek. Resize and digital processing by Masur. – Stanislaw Lem 2.jpg, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=915059

Here’s how the apparently impossible can be achieved. Hotel Infinity is a hotel with an infinite number of rooms, each room is occupied by a single guest, and none of the guests is willing to share their room with anyone else. A new guest arrives asking for a room for the night and the Hotel Manager is happy to oblige. In order to accommodate the new guest, the Hotel Manager can simply move Guest 1 from Room 1 into Room 2, move Guest 2 from Room 2 into Room 3 and so on. Then place the new guest into Room 1. In Hotel Infinity there will always be new rooms to move guests into.

Similarly, one can accommodate, say, five new guests. Move Guest 1 from Room 1 into Room 6 (1 plus 5), move Guest 2 from Room 2 into Room 7 (2 plus 5) and so on. Therefore one can accommodate g new guests by moving Guest 1 into Room 1 + g, move Guest 2 into Room 2 + g and so on. The Hotel Manager can, therefore, accommodate any finite number of new guests since g can be as large as you like.

What is more amazing is that one can also accommodate an infinite number of new guests. Simply move Guest 1 from Room 1 into Room 2, move Guest 2 from Room 2 into Room 4, move Guest 3 from Room 3 into Room 6 and so on. All the guests are now in the even numbered rooms and all the odd numbered rooms are empty. Since there are an infinite number of odd numbers, one can accommodate an infinite number of new guests.

What about an infinite number of infinite sets of new guests? Easy. First of all, do what you did in the previous case and free up all the odd numbered rooms again. Now take a look at the array in my illustration at the beginning of this post depicting an infinite number of guests in each of Bus 1, Bus 2, Bus 3 and so on, in fact an infinite number of buses. By following the arrows you can see that one can select all of the passengers on all of the buses one at a time and place them in those odd numbered rooms without overlooking any of the bus passengers. These infinities are called countable or denumerable infinities.


CANTOR’S DIAGONAL DIAGRAM IN HIS ARGUMENT FOR THE EXISTENCE OF UNCOUNTABLE SETS
By Cronholm144 – Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=2206106

THE MOST EXTRAORDINARY THING OF ALL

Despite all these ingenious methods of accommodating larger and larger sets of infinities in Hotel Infinity, sometimes you can’t accommodate everyone new. The infinities we’ve been talking about are countable infinities. However it has been proven with mathematical rigour (over a century ago, by Georg Cantor), that some infinities are uncountable, and are larger than countable infinities, as bizarre as that sounds. The number of Natural Numbers (one, two, three etc.), designated aleph null, is countably infinite. However the number of Real Numbers, designated aleph one, is not countable, so there are more Real Numbers than there are Natural Numbers even though the Natural Numbers are infinite. If the size of the set of new guests was the same size as the size of the Real Number set then some of the guests would be out of luck.

NEXT POST – Pythagorean Infinity

POSTSCRIPT: More Advanced Accommodation Strategies:

There is a second way of accommodating an infinite number of infinite sets – The Prime Powers Method. Start with powers of two. Before the buses arrive rearrange the occupants who are already in the hotel. Put the guest from Room 1 into the room numbered two to the power 1 (Room 2). Put the guest from Room 2 into the room numbered two to the power 2 (Room 4). Put the guest from Room 3 into the room numbered two to the power 3 (Room 8). Continue so that all infinite guests are now in the infinite rooms numbered with the infinite powers of two (two is the first prime number). When Bus 1 arrives place its infinite occupants into the rooms numbered with the infinite powers of three (three is the second prime number). Place the infinite occupants of Bus 2 into the rooms numbered with the infinite powers of five (the third prime), and place the infinite occupants of Bus 3 into the rooms numbered with the infinite powers of seven (the fourth prime). Continue doing this forever, for as Euclid first proved about 2300 years ago the number of primes is infinite. It is because we’re using primes that none of these infinite prime power sets overlap. Furthermore, this way you also end up with an infinite number of unoccupied rooms.

We used the Prime Powers Method to accommodate two layers of infinity (an infinite number of infinite sets). There are at least four other different ways to accommodate these two layers of infinity: The Prime Factorizations Method, The Interleaving Method, The Triangular Number Method and The Arbitrary Enumeration Method. The nice thing is that one can also use those four methods to accommodate three, four, five, and in fact an infinite number of levels of infinity.

The Prime Factorizations Method, for example, involves infinite powers of prime numbers. All you need to do is put a new guest who was sitting in seat s of bus b into the room designated by the product of 2 (the first prime) to the power s and 3 (the second prime) to the power b. For example a new guest sitting in seat 4 of bus 5 would go into room 3888 (16 x 243). The value of b would be zero for the people already in the hotel. Since every number has a unique prime factorization, this solution even leaves some rooms empty. Here is a link to more details – https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel .

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