# Hilbert’s Hotel shows why some infinities are bigger than others

Here is an entertaining video that has a nice explanation of a seeming paradox about infinities.

In 1924, the German mathematician David Hilbert raised a peculiar and seemingly paradoxical question: are some infinities bigger than others? The answer he arrived at – yes, actually – might have been impenetrable to non-mathematicians if not for the thought experiment he devised involving a hotel with an infinite number rooms. This video from the Australian filmmaker and educator Derek Muller builds Hilbert’s ‘infinite hotel’ and populates it with some strange, fuzzy creatures to demonstrate how the mathematician arrived at his groundbreaking conclusion, and touches on the real-world implications of his discovery.

Now, I don’t remember much from my higher education, which finished with a BA degree, yet I do remember learning that Georg Cantor first demonstrated the cardinality of infinite sets, and the diagonalization proof. (I just knew that computer science degree would come in handy some day!) So I’m unclear what Hilbert added to it. Since so many people reading this are smarter than me, someone fill me, please.

2. consciousness razor says

Right, I’m with you. I see Hilbert’s Hotel as a nice way to portray or diagnose some of the weird properties of infinite sets — including countably infinite ones, which is often the assumption made about the hotel, not that its rooms are uncountable.

But as you said, the fact that there are uncountable sets with a larger cardinality than the natural numbers was proven much earlier by Cantor. (I don’t know, but I wouldn’t be surprised if Hilbert also came up with some proofs of this too.)

Maybe it can help to “show” something about that, although it really has to do with the differences between finite and infinite sets. But anyway, it’s not demonstrating “why,” because it’s just a paradoxical thing and not a proof.

3. JM says

The argument they give in the video is just a different way of explaining Cantor’s proof also.

Looking at his video’s I’m guessing this ended with the name Hilbert Hotel because Veritasium did a video talking about Hilbert’s attempt to formalize mathematics and why it failed. The Hilbert Hotel video is a more detailed explanation of something mentioned in passing from this: https://www.youtube.com/watch?v=HeQX2HjkcNo. The Hilbert Hotel came out first but it’s explaining something that seems paradoxical and was likely done because of the second.

4. xohjoh2n says

The problem with Hilbert’s Hotel is that it’s dead easy to get a reservation, but it takes *forever* to check in.

(Hilbert introduced the Hotel as a means of teaching Cantor’s theories on infinite sets and make some of the consequences, which are provably true yet quite odd, more accessible. That’s not related to his formalism work which was shot through by Gödel.)

5. says

Maybe somebody can help me with this. There is something about this “thought experiment” that I don’t understand. I’m not even sure about the simplest solution of the simplest case: the case in which one guest-candidate shows up, and the existing guests are all asked to move up one room number, emptying Room One (if I might call it that) for the new guest.

This is the only way in which I have been able to understand the “Room One” solution to the Infinite Hotel experiment. The hotel is described as having an infinite number of rooms, each with a guest in it. When all the guests are asked to move to a room of one higher number, it seems to me that nothing has necessarily changed in substance; it is still a hotel with an infinite number of rooms, each with a guest in it--and with an empty twenty-by-thirty-foot box at the near end. The “box” is, after all, described in the thought experiment as a hotel room, not as a room intended for the arbitrary ejection of legitimate guests.

What I am getting at, I suppose, is the notion that the hotel as described consists of units--but it is only in the conceit of an observer that the hotel would be said to consist of an infinite number of rooms, occupied each with one of an infinite number of guests. It would be just as correct to describe the hotel as an infinite number of occupied single-occupancy guest rooms. Would the “one extra guest to be accommodated” scenario be satisfied by a person stating that the hotel company could simply construct a minimally-sufficient twenty-by-thirty-foot utility-connected box at the hotel’s near end?

I know “making another hotel room” is a rather silly solution, but then so is the notion that one could make an infinite number of people change hotel rooms to accommodate a single person. Moreover, the question at hand is not a matter of real-world practicability, but rather a matter of how the Infinite Hotel experiment could be addressed without arbitrarily changing its conditions. When the standard solution to the “One Extra Guest” scenario is used, Room One--as I stated above--has been changed into as empty a twenty-by-thirty-foot box as my hypothetical constructed one. Each of them functions as a hotel room only insofar as we decide to call them hotel rooms, and place the extra guest in them. Since Room One now has no occupant, yet a description of the condition of the hotel--absent the emptied room--is the same as before, are we not driven to conclude that following the rules of the experiment leads not to its analysis, but to its alteration? How can an ostensible solution to a thought experiment stand, when it is logically indistinguishable from an arbitrary alteration of the experiment’s conditions?

When Room One is empty, what makes it a hotel room other than our conceit to call it so? And what is it other than our conceit that would allow us to imagine even as much as One Extra Guest? It would be as plausible to maintain that no such guest-candidate could exist; after all, the infinitude of guests (and of rooms for them) posited by the experiment would seem to leave no potential guest-candidates available--unless we decide that such people are going to exist. And that is the crux of the matter. We are deciding that certain things exist. We are deciding that our calling things by certain names makes them so.

Why not make the thought experiment about filling up an infinite number of infinite hotels? After all, we have decided that there is an infinity of guests, have we not? We have decided that the emptying and filling of hotel rooms can be done instantaneously and for arbitrary reasons. Just because we have to move someone out of one room in order to fill up another doesn’t mean that the first room can’t be immediately occupied by still another guest, a person’s whose absence elsewhere can be filled by yet another. Indeed, why not postulate an infinite number of infinite hotels packed to their limits by a single guest?

I suppose it’s just the “infinity” part of this topic that’s throwing me off, but then, it seems to be all about infinity (and, maybe, about good mathematical practices that I’ll never grasp.) At the risk of being tiresome, however, I feel compelled to ask again: How can an ostensible solution to a thought experiment stand, when it is logically indistinguishable from an arbitrary alteration of the experiment’s conditions?