Every Hotel is Hilbert's Hotel
On taming The Infinite
Once upon a time,
, who is sound on almost everything except his dogged skepticism of mathematics (and bitcoin, apparently) published a 'stack attacking infinity again. Yes, it was a year back, but I am a lazy writer.
Since there is nothing paradoxical about any of these, I decided to publish a Mathematician’s Defense of at least one of the three alleged paradoxes, namely Hilbert’s Hotel. I picked Hilbert’s Hotel (I would love to abbreviate it but the initials are quite unfortunate) because Banach-Tarski would involve delving into Measure Theory and the mind-bending concept of a non-measurable set, and Gabriel’s horn seemed too trivial to tackle (real-life paint is not infinitesimally thin and therefore there is no paradox). Additionally, the following framing of Hilbert’s Hotel is, as far as I can tell, original to me.
My simple claim is that there is nothing (too) magical about Hilbert’s Hotel, that indeed any ordinary hotel can be turned into Hilbert’s Hotel. My hope is to make The Infinite a more normal part of people’s concept-space and bring it down a few notches from the mystical and ethereal pedestal upon which it is so often placed. If you are already familiar with Hilbert’s Hotel, you might want to skip ahead until you see the subscribe button after the share button below.
The set-up for the Hilbert’s Hotel thought experiment is as follows: Hilbert’s Hotel has infinitely many rooms; this leads to the weird property that even when full, it can accommodate additional guests, indeed it can accommodate infinitely many guests, provided they come through the door in a single file (i.e they form a countable set). How does this work, you may ask? It’s quite simple - suppose you ring the bell in the lobby to summon the concierge and demand a room despite having seen the ‘No Vacancy’ neon sign flashing brightly on your way in (I may be thinking of Hilbert’s Motel). The concierge, nevertheless, is eager to please, and asks you to move into Room No. 1.
If you’re worried about the present occupant(s) of Room No. 1 and what he may do to you, do not fret. They will be moved into Room No. 2! Now you may scoff and exclaim that we’ve merely pushed the problem back one step, but no, the occupant(s) of Room No. 2 will be moved to Room No. 3, whose occupant(s) will in turn be moved to Room No. 4, and so on. In an ordinary hotel, with say 500 rooms, the initial occupant(s) of Room No. 500 would have nowhere to go (we’ll come back to this later). However, since Hilbert’s Hotel has an infinite number of rooms, there is no final room that we have to worry about! Every set of occupants merely grumbles for a bit as they move their bags around, but they have a place to stay regardless.
Now suppose infinitely many guests show up, all wanting their own rooms, and conveniently wearing lapel pins with their numbers on them. No problem! We simply move the occupant(s) of Room No. 2 to Room No. 4, the occupant(s) of Room No. 3 to Room No. 6, the occupant(s) of Room No. 4 to Room No. 8, the occupant(s) of Room No. 5 to Room No. 10, and so on, moving every occupant of a prime-numbered room to the room that has twice the number of their original room, and moving the occupants of those rooms in turn to the room with twice the number of their original rooms. You may want to pause for a second to understand why there is no clash in this arrangement; the proof is left as an exercise to the reader.
As a consequence of this merry mixer, all the prime-numbered rooms are now vacant, and our infinitely many guests proceed to fill them up one at a time. Everyone is satisfied! I leave it to Euclid to show that there are indeed infinitely many empty rooms under this schema.
Can Hilbert’s Hotel accommodate arbitrarily large sets of new guests when full? Surprisingly, no. The clever tricks that the concierge plays to satisfy his customers only work as long as the set of guests is countable. If an uncountably infinite number of guests arrive, then there is nothing the concierge can do to accommodate them all, even if the hotel is completely empty. But that is a discussion for a 'stack unto itself on the general concept of Diagonal Arguments.
Now that you are familiar with the Hotel, what of my claim that every hotel can work like this? An astute reader may notice that all the clever combinatorial tricks strictly require that there is no upper bound on the number of rooms. How could your local neighbourhood Hilton possibly satisfy such a ridiculous requirement? Permit me to invoke Huemer’s notions of extensive and intensive magnitudes with regard to infinities, and posit that time is - both forward and backward but for our purposes forward alone will do - infinite. Thus what we have are not just finitely many rooms, but finitely many (and at least one) rooms(s) every day, forever, until the (non-existent) end of time.
In case you are unfamiliar with the distinction Huemer draws between extensive and intensive magnitudes, you should go read Approaching Infinity (after finishing this essay). It has been a while since I read the book myself, so I pass the mic to ChatGPT to summarise the distinction.
Extensive magnitudes are those that can be divided into smaller parts, where the whole is the sum of its parts (e.g., length, area, or volume). These can conceptually be infinite because they can be continuously subdivided and summed.
Intensive magnitudes refer to properties that cannot be subdivided in this way, such as temperature, density, or brightness. These magnitudes represent the degree of something at a point rather than over a region, and hence, they cannot be infinite, as they do not accumulate by summing over parts.
In particular, time is an extensive magnitude. Now, the astute reader may have already figured out my approach. Since we now have finitely many rooms over an infinitely long period, we can index the rooms not merely by number, but also by day (the specific unit of time chosen is irrelevant, we could have picked years or even centuries, as long as the forward time horizon is infinite). Suppose we have N rooms, then we label rooms on day 1 from 1 through N, on day 2 from N+1 through 2N, and so on. Each (day, room) is assigned a unique natural number, and conversely each natural number corresponds to a particular room on a particular day.
Now that we have an index of countably infinite rooms, we can begin filling them up with our infinite guests, who also conveniently come with natural numbers attached to them. However, they do not care about their associated numbers and we are permitted to rearrange their faces, and give them all another name. Each guest must be assigned one room on some day, but importantly they do not have a preference for timing or location (in the classic Hotel, guests similarly did not care about which room they were assigned). This can easily be done by matching up the guests’ indices with the corresponding (day, room) under our aforementioned labelling schema. Now, as a second batch of infinitely many guests demand accommodation, we can repeat the classic Hilbertian prime-switching technique to satisfy them all.
One further assumption that can be relaxed is the strict one-day-per-guest assumption, as long as each guest only wants to stay for a finite number of days. Simply decompose each guest into a number of guests who all stay for one day each (do not try this at home), and relabel this new countably infinite list by starting from the copies of Guest 1 and proceeding with the rest. This would be akin to guests at the classic Hilbert’s Hotel demanding multiple rooms, which is identical to the case of multiple guests each demanding one room each. The mathematical logic is ambivalent to such quirks. De gustibus non est disputandum.
Am I cheating, or begging the question? I don’t think so. Patterson’s invocation of Hilbert’s Hotel as “paradoxical” is meant to invoke some intuitive sense of contradiction, and thereby convince the reader that the concept of mathematical infinity is blasphemous. I am merely trying to assuage the intuition and demonstrate that what prima facie seems like an affront to all common sense is in actuality quite tame. Most mathematics is like this, as the field itself is merely a formal, rigorous reconstruction of our eidetic intuitions. Take, for instance, the definition of (uniform) continuity of functions between Euclidean spaces:
While the formal definition may seem abstruse and arcane, the eidetic intuition that it captures is quite simple - small changes in the domain should correspond to small changes in the range. As undergrads learn to “see” mathematics, the scales fall away from their eyes and the formalism aligns with their intuition.
Thus, through a simple reframing of Hilbert’s Hotel by unfolding it over time, its ephemeral properties are collapsed into tamer, realistic ones that every hotel can potentially satisfy. The “paradox” as it were vanishes. Here I am relying on the assumption that, while a hotel with infinitely many rooms strains one’s intuition, an infinite time horizon does not, at least not in the same way. This is evidenced by all the fairy tales that end with a happily ever after.