It’s down at the end of Infinite Street… Hilbert Hotel

March 11, 2025 6:33 AM

A six minute video illustrating Hilbert’s paradox of the Grand Hotel with cute blobby monsters.
posted by Lemkin (55 comments total)

Hilbert clearly didn't check Trivago.
posted by fairmettle at 6:47 AM on March 11 [1 favorite]

Hilbert's Hotel is kind of well known in philosophy of religion because William Lane Craig used it in arguing for his version of the Kalam Cosmological Argument. He used it to say that the counterintuitive properties of transfinite numbers mean that it is impossible to have an infinite past. This was problematic because Craig wanted to be able to subtract guests which leads to complications (in most transfinite math you can't just subtract infinites of different cardinalities). Most of the current argument around the Kalam lately has focused on a different style of argument about infinites from the work of José Benardete, the Grim Reaper paradox. In this one you have a grim reaper that will kill Fred if he is alive at 1:00, a second that will kill him if he is alive at 12:30, a third that will kill him if he is alive at 12:15, and so on to infinitely many reapers. The problem being that you can't find which reaper would kill him but he must die.

It's all interesting but it does get you very much into counting angels on the heads of a pin territory.
posted by graymouser at 7:14 AM on March 11 [2 favorites]

This reminds me of those escalating childhood fights:
You stink.
Yeah? Well, you stink times two.
You stink times a hundred.
You stink times a googol.
You stink times a googolplex.
You stink times infinity.
Times infinity plus one!
posted by dances_with_sneetches at 7:24 AM on March 11 [1 favorite]

I'm sorry, but that's complete nonsense, isn't it?

If the infinity bus containing an infinite number of ABAB people includes every possible juxtaposition of As and Bs, then the new name you come up with by switching the diagonal letters must already be on the bus, right? If there are an infinite number of juxtapositions, you can't just invent one more.

I think the problem comes with the idea of there being *more* infinity. If you've got an infinite number of people occupying an infinite number of hotel rooms, there is no one left to show and ask for more hotel rooms -- that is, if there are people left over, then the number of people in the hotel is necessarily finite.

I need a nap now.
posted by Ben Trismegistus at 7:38 AM on March 11 [5 favorites]

If the infinity bus containing an infinite number of ABAB people includes every possible juxtaposition of As and Bs, then the new name you come up with by switching the diagonal letters must already be on the bus, right?

So it would seem to this layman. But I assume the Math! people will be along to resolve the difficulty.
posted by Lemkin at 7:51 AM on March 11 [3 favorites]

The point is that the person is on the bus, but there is no hotel room reserved for them. No matter how you try to make the list, someone on the bus will not be able to fit into the hotel.
posted by eruonna at 8:04 AM on March 11 [3 favorites]

My mathematician spouse has a presentation that she's given at a number of Nerd Nite events now, which is basically a lot of non-intuitive stuff about Infinity, and includes a good amount about the Hilbert Hotel and Cantor's Diagonal Argument. and I'll say that for this layman, she's able to explain the Diagonal Argument well enough that it will always make sense to me in the moment and then reliably fall out of my head later. But also I know that it was a revolutionary idea that took the Math community a long time to fully accept as well.
posted by Navelgazer at 8:18 AM on March 11 [4 favorites]

the number of rooms at the Hilbert hotel is infinite, sure, but it is countably infinite [5:06]

oh, you want Laplace, next door [ijmaa]
posted by HearHere at 8:39 AM on March 11 [3 favorites]

If the infinity bus containing an infinite number of ABAB people includes every possible juxtaposition of As and Bs, then the new name you come up with by switching the diagonal letters must already be on the bus, right?

So basically you can have either

1) an unlimited amount of people that you can each assign a unique infinitely long string of A/Bs to and line them up and count.

or

2) an unlimited amount of people similarly uniquely assigned and, for every infinite A/B string, one person was definitely assigned that string.

but not both at the same time.

Another related result is (simplifying): suppose you line up every possible finitely-large computer program. Some of them halt and crash, but the ones that don't produce a, possibly infinitely long, number between 0 and 1 (you can run the program for as long as you like and successively closer approximations; 1/2 would just be 0.5000...). Each one can be assigned a unique number (eg just write down numerical value of all the bytes of each program). Now, suppose that for every number in 0..1 there's a program that outputs it (or successively closer approximations thereto).

But now, define a number that is the fraction of computer programs that halt. That's between 0 and 1, so there must be a program that calculates it. Except that's equivalent to solving the Halting Problem, which is known to be impossible (via a diagonalization argument!). So there must be real numbers that are not computable at all, and in fact almost all numbers are not computable, because programs are of course countable and reals aren't.
posted by BungaDunga at 8:41 AM on March 11 [5 favorites]

The point is that the person is on the bus, but there is no hotel room reserved for them. No matter how you try to make the list, someone on the bus will not be able to fit into the hotel.

Why would reservations matter if there are an infinite number of rooms? And if there is someone who doesn't fit, the hotel is by definition finite.
posted by Ben Trismegistus at 8:43 AM on March 11

That would depend on how you define finite. Is the set of positive integers finite? That is the size of the hotel.

The mathematical definition of finite that is relevant here would be that a set is finite is it has a number of elements equal to some natural number, and a set is infinite if it is not finite. Under this definition, the hotel is infinite. The set of guests on the party bus is also infinite. But there are always more guests on the bus than there are rooms in the hotel.
posted by eruonna at 8:48 AM on March 11 [2 favorites]

One thing to note is that, even if you find the idea of infinities that are in a sense "larger" than each other either confusing or unlikely, if you take it as true you get an entire panoply of mathematics that remains consistent if wildly weird and unintuitive.

It all, in other words, works, and doesn't produce any actual contradictions. This isn't a little cul-de-sac of mathematical thought, but a foundation that's produced an enormous amount of perfectly good mathematics.
posted by BungaDunga at 8:50 AM on March 11 [4 favorites]

"Infinity" does not mean what most of us think it means.
posted by grumpybear69 at 8:55 AM on March 11

And if there is someone who doesn't fit, the hotel is by definition finite.

That's assuming what is to be proved, as in: it turns out this is not true. There are sets of, as it were, "pigeons" that are too large to fit into other specific sets of "holes".

There is a definition of infinity that works like you are thinking: absolute infinity, Ω. But the infinities people are familiar with (that of the natural numbers, etc) have different properties.
posted by BungaDunga at 8:55 AM on March 11 [2 favorites]

another way to think about this is that different sets have different structures (eg natural numbers, real numbers, etc). when you start talking about "infinities" you are actually talking about a specific structure or property of particular sets.

It turns out that certain sets are similar in this way (natural numbers, even numbers, odd numbers, primes all have the same "infinite cardinality") so you can lump them together. But now you've locked yourself into a particular (but very natural) definition of cardinality, suddenly you find that certain other sets must be qualitatively different, they must have a different cardinality.
posted by BungaDunga at 9:00 AM on March 11 [2 favorites]

I think the mistake was putting numbers on the hotel rooms. If you'd simply labeled them AA.... BA.... AB... BB... and so on you'd have enough rooms for the last nonsense bus full of people, AND you can tell them to simply walk to the room with their name on the door.
posted by the antecedent of that pronoun at 9:03 AM on March 11 [2 favorites]

(though I gather there would be some theoretical problem with giving them accurate directions, such there not being enough sequences of instructions like "go up an infinite number of flights of stairs, then take the infinity'th left turn, then go to the infinity'th room on the right in that hall" to guide them all to those named rooms)
posted by the antecedent of that pronoun at 9:06 AM on March 11 [1 favorite]

If you want to muddy your brain further: the countable infinity of the natural numbers is smaller than the infinity of the real numbers. So the obvious question is, is there an infinity bigger than the natural numbers and smaller than the real numbers? It turns out that we can't prove it. However, we can prove we can't prove it. We also can prove that we can't disprove it. Welcome to the shambling hell of the continuum hypothesis.
posted by phooky at 9:10 AM on March 11 [10 favorites]

I'm not sure that adding Hilbert's Hotel to the the proof that the cardinality of the reals is greater than the cardinality of the integers makes anything simpler.

I wish he'd made explicit that in order to accommodate a guest you must assign them to a room with a specific number. You must be able to say "Go to room X". You can't say "Keep walking until you find an empty room", because they never will. Or, to put it another way, if you are moving guests around or trying to accommodate a new one, every guest must only need to walk a finite distance to get to their room. It's obvious enough when you state it that way, but not everyone watching this is going to have been exposed to this stuff, so it's worth being explicit.
posted by It's Never Lurgi at 9:11 AM on March 11 [3 favorites]

I gather there would be some theoretical problem with giving them accurate directions

My favorite comment on the video was “Imagine kicking an infinite number of people out of their hotel rooms to accommodate a single extra guest”.
posted by Lemkin at 9:12 AM on March 11 [1 favorite]

grumpybear69: "Infinity" does not mean what most of us think it means.

BungaDunga: There is a definition of infinity that works like you are thinking: absolute infinity, Ω. But the infinities people are familiar with (that of the natural numbers, etc) have different properties.

I think this may be my issue. My very basic understanding of the idea of infinite numbers is that it is never-ending and contains all possible values. So even though there is an infinite number of fractions in between the infinite number of integers, that wouldn't mean that there are "more" fractions than integers, because both numbers are infinite.

In my tiny brain, it seems like, if anything can be said to be "bigger" than something else, that something else can't be infinite. But I concede that I don't really have any idea what I'm talking about.
posted by Ben Trismegistus at 9:14 AM on March 11 [1 favorite]

And I thought checking in at Caesars on a Friday afternoon was a long line.
posted by Lemkin at 9:15 AM on March 11 [1 favorite]

MetaFilter: I concede that I don't really have any idea what I'm talking about
posted by Lemkin at 9:16 AM on March 11 [6 favorites]

I think the mistake was putting numbers on the hotel rooms. If you'd simply labeled them AA.... BA.... AB... BB... and so on you'd have enough rooms for the last nonsense bus full of people, AND you can tell them to simply walk to the room with their name on the door.

But this does mean you are adding more rooms. One of the earlier buses would not fill this hotel.
posted by eruonna at 9:16 AM on March 11

And I thought checking in at Caesars on a Friday afternoon was a long line.

No, the long line is something else.
posted by eruonna at 9:17 AM on March 11 [6 favorites]

Why would reservations matter if there are an infinite number of rooms? And if there is someone who doesn't fit, the hotel is by definition finite.

This is the thing: There are different sizes of infinity. "Countably Infinite" (e.g. the counting numbers) is the lowest order of that. Reals are "Uncountably Infinite," which is a higher order of that.
posted by Navelgazer at 9:25 AM on March 11 [1 favorite]

My very basic understanding of the idea of infinite numbers is that it is never-ending and contains all possible values.

Yeah, that's what most people think. Honestly, before set theory came about, mathematicians thought of infinity as a limit (i.e. something you can approach, but not reach) and not as an actual "thing" about which you can reason.

Cantor showed that you can treat infinity as an actual thing and that leads to a huge amount of fun. But (and this is important), while the fun is extremely counter-intuitive, it's not contradictory.

The thing is, you already know that infinity doesn't have to contain everything. You agree that there are an infinite number of positive numbers, right? But, that set of numbers doesn't contain the negative numbers, so it doesn't have everything. The set of all prime numbers is infinite, but it definitely doesn't contain everything. If the Hilbert Hotel had rooms 1-10 temporarily closed for renovations, it would still have an infinite number of rooms.

At this point you might stop and consider if there is a better definition of "infinite" than "contains everything" or some vague "goes on forever... and stuff". There is.

So how can one infinity be bigger than another? Once you get a more nuanced definition of infinity, you might well ask the opposite question - is there any way to tell if two different infinities (say, the number of integers and the number of prime numbers) are equal? It turns out that there is a very reasonable way to do this ("reasonable" because it works in a very intuitive way for boring old finite numbers and extending it to infinite numbers doesn't break anything) and that's how you get infinities that are not equal to each other.
posted by It's Never Lurgi at 9:28 AM on March 11 [8 favorites]

Cantor showed that you can treat infinity as an actual thing and that leads to a huge amount of fun. But

I never have the patience to make it to the end of these things. And anyway, the flaw (the problem?) is with the math (or perhaps the logic) not with infinity, which allows for everything. In other words, Cantor's wrong, this isn't fun. Everything will be okay.
posted by philip-random at 9:35 AM on March 11

If the Hilbert Hotel had rooms 1-10 temporarily closed for renovations, it would still have an infinite number of rooms.

And presumably an infinite number of little soap bars.
posted by Lemkin at 9:37 AM on March 11

then the new name you come up with by switching the diagonal letters must already be on the bus, right?

They're on the bus, but they're not on the manifest of people occupying rooms. If you put them in the first room, and then change all the letters on the diagonal again, you generate another new name that is on the bus but isn't yet assigned a room.

The math interpretation is that there are infinitely more "irrational" numbers, whose decimal representations don't end and never repeat, than there are "rational" numbers, whose decimal representations do eventually repeat. (A "terminating decimal" ends in a repeating sequence of zeros.) A rational number gets its name because it can always be represented as a ratio of two natural numbers.

You can "count" the rational numbers, using the diagonal zigzag approach in the video. The set of all natural numbers is the same size as the set of all even numbers, which is the same size as the set of all rational fractions.

But if you try to "count" the real numbers, by assigning each of them a position in a list, you'll always omit infinitely many real numbers.
posted by fantabulous timewaster at 9:39 AM on March 11 [5 favorites]

This 1985 episode of NOVA gets into Russell's paradox in set theory.
posted by neuron at 9:40 AM on March 11 [1 favorite]

But this does mean you are adding more rooms. One of the earlier buses would not fill this hotel.

Yes, true. and I'm not sure the "how to move rooms when a countably-infinite bus shows up" rules for numbered rooms translate to named rooms. (say the uncountably infinite rooms were already full, and a countably-infinite bus shows up...) What's GOOD about this case is, assuming you can work that out, when you ask the INFINITE number of existing guests to change rooms to make space for the INFINITE new guests, I think that the total fraction of existing guests who have to move is 0.

Of course if another ABBA bus comes you can probably figure out a re-naming but then everybody'll have to move.
posted by the antecedent of that pronoun at 9:45 AM on March 11

You stink times infinity.

"Yeah well you stink an infinite number of buses all full of infinite people all with infinite names of every combination of ABBA... plus one."
posted by Flaffigan at 9:47 AM on March 11 [1 favorite]

For everyone finding this bewildering, you're not alone. In the late 19th and early 20th century this was kind of the mathematical hangup that screwed with everyone's head, in a few different guises. Cantor's discovery was upsetting, incendiary, and widely rejected for a good long while. A similar argument in self-reference presented by Bertrand Russell to Gottlob Frege completely demolished the latter's attempts to systematize set theory, to the extent that he tried to actually suppress publication the second volume of his own work (but was overruled by his publishers, and settled for adding an appendix which explained why the entire work was wrong and bad). And of course, the same fundamental construction of a self-referential counterexample which was at the core of Cantor's Diagonal argument and Russell's Paradox would likewise occur in Gödel's Incompleteness Theorem, and several decades later recurred as the Halting Problem in computer science.
posted by jackbishop at 9:58 AM on March 11 [9 favorites]

★☆☆☆☆ INFINITELY AWFUL!
Arrived around 1AM after a horrible flight on Zeno Air. Hotel management assured me they'd have a room but when I got there, they told me I had to wait for housekeeping to clean my room. FINALLY, after something like 10^2342323390 hours, they handed me the key card (which was the size of one of those Publishers' Clearing House checks, WTF???). Thought I could get some sleep, but NO! Apparently a massive tour bus showed up and I had to move to a room TWICE AS FAR from the front desk. Management provided a luggage cart, but it still took a ridiculous amount of time to find my new room. Needless to say, the view was terrible, looking out over an infinite parking lot with infinite sulfur lighting that kept me up another whole night. Then another bus showed up, this time with no seats and packed like a sardine can. Well, I've been traveling on business for years and I know when it's time to leave. I checked out early, but the desk clerk was rushing around doing other things. I threw my keys down on the desk, walked out, and called my credit card company for a chargeback. I'm still waiting!!!
posted by PlusDistance at 10:05 AM on March 11 [13 favorites]

If you want to specify what comes after infinity — by insisting that none of the guests should change rooms to accommodate a new arrival — you need the transfinite ordinal numbers.
posted by fantabulous timewaster at 10:18 AM on March 11 [2 favorites]

In my tiny brain, it seems like, if anything can be said to be "bigger" than something else, that something else can't be infinite.

Learning that there are different cardinalities of sets (and grokking why it must be true) is one of the three most important & vivid memories in my life, and permanently shifted how I view the world.

The other two memories are grokking [to a certain extent] the incompleteness theorem; and my very first memory at age 3: sitting in the back seat of a car while my father showed my uncle the nearly completed house he had built (with help only for pouring the foundation & some of the roof framing) and my uncle saying "well, that is really something".
posted by lastobelus at 12:51 PM on March 11 [3 favorites]

Related Stand-up Maths video: An infinite number of $1 bills and an infinite number of $20 bills would be worth the same.
posted by Pendragon at 1:13 PM on March 11 [2 favorites]

I learned about it the following way:

Think of an infinite line.

Now think of an infinite plane.

An infinite number of infinite lines can fit in an infinite plane.

Not a single infinite plane can fit in an infinite line.
posted by kyrademon at 1:54 PM on March 11 [1 favorite]

I am no math scientist, but isn't the problem that "infinity" is not actually...real? That is, the human mind actually can't conceive of infinity. It gets to how we live in the 4th dimension of time, and with time you can just keep adding. So "infinity" is more like X in an equation, which is singular.

So "infinite guests in infinite rooms" as a basis for a puzzle is a non-starter, because no one knows what that means. Our brains don't work that way.
posted by zardoz at 1:57 PM on March 11 [1 favorite]

Unfortunately I think they can. At least, I think wikipedia thinks they can.

Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space

You can do this when you "interleave the letters of the name of the bus with the letters in name of the passenger", you've got a bunch of infinitely long and still distinct names for every passenger on every bus, so that they can all fit into the rooms of your hotel with names on the door (instead of integers)
posted by the antecedent of that pronoun at 1:59 PM on March 11 [3 favorites]

An infinite number of $1 bills and an infinite number of $20 bills would be worth the same

But an infinite number of nickels would be worth more than an infinite number of dimes, because they’re bigger.
posted by Lemkin at 2:00 PM on March 11

The cardinality of a line and a plane is indeed the same. This stuff is weird.

One visual way to see how you can put a plane and a line into exact correspondence is with a space-filling curve.
posted by BungaDunga at 2:11 PM on March 11 [1 favorite]

I am no math scientist, but isn't the problem that "infinity" is not actually...real? That is, the human mind actually can't conceive of infinity.

I'm not sure the human mind can conceive of 734,981,250,115,004,992 either, but it's a pretty standard number.

What is "real"? Is 5 real? What color is it? How much does it weigh? Fine, I'll grant you the positive numbers. Is -3 real? What about 3/2? What about sqrt(7)? Pi? What about sqrt(-1)?

We dealt with infinity before as a limit (rational numbers repeat forever, integer go on forever), but not as a thing. The great insight of Cantor is that you can treat infinity as just another quantity as long as you follow the rules and are prepared to adjust your intuitions.
posted by It's Never Lurgi at 2:25 PM on March 11 [2 favorites]

The problem is that people think of infinity as a really big number, but it's not a number at all.
posted by Pendragon at 2:29 PM on March 11

MetaFilter: I concede that I don't really have any idea what I'm talking about

Look, infinitely long hotel hallways and airplanes that only ever fly halfway to their destination are one thing - but this is just unrealistic.
posted by nickmark at 3:11 PM on March 11

the human mind can't actually conceive of infinity

My mind can't even conceive of moderately large numbers like, say, 73. I can't hold some kind of image of 73 in my mind that reveals how it is different from 72 or 74 or any other number. So inconceivability is not just a feature of infinity ---- it's a feature of any number I can't count on my fingers and toes.

What I can do with 73 is operate on it - I know how to do operations on 73 and all the other numbers. So I can tell 73 is larger than 72 and smaller than 74 by looking first at the leftmost digits - they're all the same -- so I look at the next digits to the right, and I see 3 is larger than 2 but smaller than 4 -- maybe I can conceive of those single-digit numbers, or maybe I just memorized those facts once.

our brains don't work that way

I wasn't born knowing how to compare large numbers. It was a skill I was taught. Likewise, people have figured out how to do operations and analyses on the various infinities, and that can be learned too.
posted by JonJacky at 3:22 PM on March 11 [2 favorites]

> "The cardinality of a line and a plane is indeed the same."

Yup, you're right.

Sorry for adding to the confusion, all.
posted by kyrademon at 5:22 PM on March 11

Yeah, coming in late, but this is where I always sort of sigh at how popular Veritasium is as a science educator, because I always sort of think his explanations sort of end up feeling like he edited out important steps while simplifying it for a mass audience. There are other science educators who do a better job of it, but I think the unfortunate part is that at least for some of these topics, you do kind of have to be willing to go down the rabbit hole a bit further than Derek does. Am I mathematician? Hah, hell no. Have I had other science educators explain it to me in a way that made the little gray cells light up and go "oh! huh, ok. Neat!" Absolutely.
posted by Kyol at 6:25 PM on March 11 [1 favorite]

The problem is that people think of infinity as a really big number, but it's not a number at all.

Yes. A good way to think of it is that it is not a point on the number line, the way that 5.4 or -2000 are. In that sense, infinity isn't "real".

But negative numbers aren't "real". Pi isn't "real". Complex numbers aren't "real". And when it comes down to it, numbers like eleven or zero aren't "real" either.

And yet, because of the unreasonable effectiveness of mathematics, there's definitely something real there.

The rest is just numerology.
posted by AlSweigart at 7:26 PM on March 11 [3 favorites]

There's a sense in which none of this is real. In other words, mathematics in its pure form is just about selecting some system and figuring out how it works. You can, for example, figure out how geometry works in non-Euclidean systems (look up Taxicab geometry some time). It's not a given that a² + b² = c² always and in all contexts.

So layperson pronunciations about what's real is a little... arbitrary... to a mathematician. What you might think of as mathematical truth is just a particular set of truths that flow from a particular set of starting points. Select some different starting points and you can still do mathematics. It just won't look so familiar.
posted by axiom at 9:13 PM on March 11 [5 favorites]

As a very non-math person, I remember quite well a specific moment when infinity became an ordinary thing for me. It's not some great insight! But the thing is that infinity sounds like it should be very special in some way. Mystical even. The human mind can't even contemplate it!

And then someone just pointed out that there must be an infinite amount of numbers because you can always add one to any number. And that's all there is to that kind of infinity, really. And all the mystery was deflated! In fact the "opposite" (not sure what to call it) then sounds absurd, as if you can always n+1 but not to 56! Or that you can always n+1 until ten trillion and after that you can't do that anymore.

This simple thing, which is so basic and obvious that nobody even bothered to point it out to me until I was well into adulthood, entirely deflated the mystery of infinity for me. Or that sort of infinity at least.
posted by Pyrogenesis at 12:03 AM on March 12

I just wanted to pop back in and say that this is fascinating. Thank you all.
posted by Ben Trismegistus at 7:14 AM on March 12

I think this may be my issue. My very basic understanding of the idea of infinite numbers is that it is never-ending and contains all possible values. So even though there is an infinite number of fractions in between the infinite number of integers, that wouldn't mean that there are "more" fractions than integers, because both numbers are infinite.

Here's what's really interesting (and something the video IMO doesn't actually explain very well, because it doesn't always connect the dots between what it's illustrating and what that means in mathematical terms): The infinite number of integers and the infinite number of fractions (or "rational numbers" is, in fact, the same. That's what was being illustrated with the spreadsheet and the criss-crossing line that they straightened out. That's how you show that fractions are countably infinite (i.e., you can line them up with counting numbers.)

It's the real numbers that are uncountably infinite. (This is what was being illustrated via the ABBA Party bus, though again that's never explicitly explained here.) There's no way to possibly line them up with the counting numbers. So that's a greater size (or order, or aleph, or whatever you want to call it) of infinity. And there are ones large than that! Like sets of real numbers (if I'm remembering this correctly.) And there may be sizes in between those, but it's been proven that it can't be proven one way or the other if those exist, so we'll literally never know!
posted by Navelgazer at 9:35 AM on March 12 [1 favorite]

The problem is that people think of infinity as a really big number, but it's not a number at all.

On the one hand yes, on the other hand ordinals behave more like finite numbers than you'd probably expect, and "infinity plus one" can actually be well-defined.
posted by BungaDunga at 1:25 PM on March 12 [2 favorites]