Hilbert’s Hotel is a thought experiment that the great mathematician David Hilbert (1862–1943) developed to help us see the “counterintuitiveness of infinity.” He asks us to imagine a hotel which is “full” — except that because it is infinite, it can always create one more room. Mathematician Marianne Freiberger explains:

Suppose that your hotel has infinitely many rooms, numbered 1, 2, 3, etc. All rooms are occupied, when a new guest arrives and asks to be put up. What do you do? It’s easy. Ask the guest in room 1 to move to room 2, the guest in room 2 to move into room 3, the guest in room 3 to move into room 4, and so on. If there were only finitely many rooms, the guest in the last room would have nowhere to go, but since there are infinitely many, everybody will find a new abode.

Marianne Freiberger, “Hilbert’s hotel” at Plus Maths (February 13, 2017)

Note, however, that all hotel guests must all move instantly and simultaneously; otherwise the moves would take an infinite amount of time. But this is a thought experiment in an ideal mathematical universe so for now we will set aside the question of whether people would or could really behave that way.

In an infinite hotel, we can easily accommodate any finite number of new guests at once:

If n new guests arrive, simply ask each existing guest to move to the room whose number is n plus the number of their existing room. As an example, if there are 8 new guests, then the guest currently in room 10 needs to move into room 10+8=18.

Marianne Freiberger, “Hilbert’s hotel” at Plus Maths (February 13, 2017)

But what if an infinite number of new guests arrives? According to Freiberger, that is a bit trickier, but it can still be done:

Suppose an infinite number of new guests arrive, forming an orderly queue outside the hotel. In this case, ask each existing guest to move into the room whose number is twice the number of their current room. So, a guest staying in room x moves to room 2x. After this manoeuvre only the even numbered rooms are occupied: rooms 2 = 2 × 1, 4 = 2 × 2, 6 = 2 × 3, and so on. The odd numbered rooms are all free, so you can put your first new guest into room 1, the second new guest into room 3, the third new guest into room 5, and so on. Everybody is happy.

Marianne Freiberger, “Hilbert’s hotel” at Plus Maths (February 13, 2017)

Senior data scientist Kasper Müller offers some more Hilbert’s Hotel teasers:

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 = 2 ⋅ 3 was sitting in bus number 7 and seat number 9. This can easily be generalized to many more layers of infinity using more primes.

Kasper Müller, “Hilbert’s Hotel – An Ingenious Explanation of Infinity” at Medium (December 30, 2021)

Müller also offers a formula for “an infinite number of ferries each carrying an infinite number of busses each having an infinite number of passengers.”

But he warns that the set of all real numbers is uncountable (so maybe the hotel would have to give up?), and notes,

Oh, and there are in fact infinitely many different kinds of infinities; the question now is, of course, which kind of infinity??

Kasper Müller, “Hilbert’s Hotel – An Ingenious Explanation of Infinity” at Medium (December 30, 2021)

Well, abstract concepts like infinity certainly help keep math interesting.

In other infinity news:

From mathematician and computer programmer Jonathan Bartlett: Yes, you can manipulate infinity—in math. The hyperreals are bigger (and smaller) than your average number — and better! Hyperreal numbers are a new type of number that was developed to simplify and rethink the way that we deal with very large and very small numbers.

and

Can we add new numbers to mathematics? We can work with hyperreal numbers using conventional methods. It could start in high school. Surprisingly, yes. It began when the guy who discovered irrational numbers was — we are told — tossed into the sea.

And from computer engineering prof Robert J. Marks: Why just anything can’t happen given an infinite sum of universes. We can see why not, using simple mathematical reasoning in this universe. It can be shown mathematically that the infinite does not exist in reality, only in our minds. Thus an infinite number of universes cannot exist.

You may also wish to read:

Gregory Chaitin on the great mathematicians, East and West. Himself a “game-changer” in mathematics, Chaitin muses on what made the great thinkers stand out. Chaitin discusses the almost supernatural awareness some mathematicians have had of the foundations of our shared reality in the mathematics of the universe.