Hilbert’s Hotel Revisited

How to accommodate infinitely many guests

Hilberts hotel. Collage by author with the help of midjourney. The building is the “Mathematisches Institut Göttingen”, where Hilbert worked most of his active life.

Making Rooms

In books about Cantor’s theory of transfinite (infinite) numbers, the example of Hilbert’s Hotel is often used to show what is possible in the realm of infinity. David Hilbert (1862–1943) was an ardent admirer of Cantor and devised the following picture to illustrate Cantor’s ideas. Wikipedia explains it:

Hilbert imagines a hypothetical hotel with rooms numbered 1, 2, 3, and so on with no upper limit. … Initially every room is occupied, and yet new visitors arrive, each expecting their own room. A normal, finite hotel could not accommodate new guests once every room is full. However, it can be shown that the existing guests and newcomers — even an infinite number of them — can each have their own room in the infinite hotel.

Our first question relates to “infinity”: What does it mean? As Cantor showed, there are infinitely many infinities. Here we regard the smallest of them, the set of natural numbers, that is ℕ = {1,2,3,…}. A set like this is called countable or denumerable, because its members may be mapped to the natural numbers. In this case, they are the natural numbers. Another countable set would be the set of all even numbers: 𝔼 = {2,4,6…}, which is countable too, although it looks smaller. A set seemingly larger than ℕ would be, for example, 𝔻 (for “double”) = {1,1,2,2,3,3…}. It is still countable. All of these sets have the same “size” or power (number of elements) or cardinality, which is usually denoted by ℵ0 (“aleph-null”). That’s the part difficult to grasp, hence Hilbert’s visualization via his hotel.

But how did the proprietor of the hotel do it? Somehow like this:

We gotta move …

All the hotel’s occupants move one room to the right, newcomer A then moves into the empty slot at the left. But there is a problem: That “moving to the right” must take place simultaneously, because a room can only hold one person (= one number) at a time. Although mathematics doesn’t know the concept of “time”, there still has to be a procedure whereby it is possible to undertake infinitely many actions at once. Is there such an algorithm? No, only the permission to do it — an axiom, allowing you to shift infinitely many occupants at the same time, but not telling you how to do it.

Before we explain, let me tell you that simple visualisations may be dangerous. Think of Einstein’s illustration of curved space creating gravitation: In the example of the dented table-cloth, he presupposes gravity to explain gravity (see hypothesis 2 of my article about gravity):

[embed]The Mystery of Gravity It remains unsolved — but there seems to be a solutionpeterripota.medium.com

And when Bertrand Russell tried to illustrate his paradox of a set that contains and doesn’t contain itself at the same time, he invented the barber who does or does not shave himself. Again, this example is faulty, see paradox 3 of my article:

[embed]What is Bertrand Russell’s Barber Doing? Still pondering: To shave or not to shave, that is the question!sciencespectrumu.com

To revert to Hilbert’s hotel, this procedure is only possible in quite a different way. In the language of set theory: We have to assume three different sets. Set #1 corresponds to Hilbert’s hotel with ℵ0 members. Set #2 contains all the newcomers, in our simple example: one member, called “A”. From these, we construct a new set. Set #3 takes one element of #2 and one element from #1, repeatedly if necessary, combining them into a new set, which again is countable and has size ℵ0, because it can be mapped one-to-one onto the natural numbers:

Combining tow sets into a third

The Axiom of Choice

But this choosing and combining has to be allowed, either by a theorem of set theory or by an axiom. In this case, it is the axiom of choice, invented by Ernst Zermelo, disputed by many mathematicians. But first, its definition (from Wikipedia, abbreviated):

The axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from each set, even if the collection is infinite.

If that’s too simple, here is another definition:

It states that for every set I and every I-indexed family (Si)i∈I of nonempty sets, there exists an I-indexed set (xi)i∈I of elements of ∪i∈ISi such that xi∈Si for every i∈I.

Indeed. So, mathematicians granted themselves permission for infinitely many, infinitely fast choices. They called this the axiom of choice, because it is supposed to enable any mathematician to select at least one element from infinitely many sets simultaneously. But does it work?

Bertrand Russell coined an analogy: for any collection of unordered pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate collection (i.e. set) of shoes; this makes it possible to define a choice function. However, for an infinite collection of unordered pairs of socks (assumed to have no distinguishing features such as being a left sock rather than a right sock), there is no natural way of choosing one sock from each pair, so one must appeal to the axiom of choice to construct the desired choice function.

But how should we do it? Take as an analogy this situation: The head of the police department of a certain district is dissatisfied with his staff. The clear-up ratio of crimes is far too low. So he devised a new law declaring: Every delict can be cleared up, it should be cleared up, it has to be cleared up, it will be cleared up. How shall we proceed? his subordinates asked. That’s your problem, he answered.

You think that’s absurd? Not in Germany. Recently, the German parliament enacted a law guaranteeing every family with little children a place in a kindergarten. How shall we do it? the managers of these institutions asked. We have no rooms, no qualified staff, no money. That’s your problem, the government said.

This is, in a nutshell, the (in)famous axiom of choice. With the help of this permit, at least two really astonishing paradoxes are possible. First, one may prove that the circumference of the circle is either 0 or ∞, but not 2π, as it should. Second, by simply turning a sphere, you are able to double its volume, hence creating a second sphere. These curious phenomena, only made possible by applying the axiom of choice, are elaborately explained here:

[embed]A Random Walk: Charting the Unmeasurable through Measure and Choice How Lebesgue’s Measure and the Axiom of Choice Shape the Boundaries of Infinitymedium.com

Then why was Zermelo so adamant in introducing and defending his axiom? Because he desperately wanted to well-order every set. What does that mean? Stay tuned for another tale of flight into the farthest figments of imagination!

Next article:

[embed]What is Real in Mathematics? What is Real in Mathematics? The natural numbers do exist. But what else? Let's recapitulate (from my previous…peterripota.medium.com