"Hilbert's Paradox of the Grand Hotel"

Apple

"Hilbert's paradox of the Grand Hotel is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and that this process may be repeated infinitely often. The idea was introduced by David Hilbert in a 1924."

"Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied. Suppose a new guest arrives and wishes to be accommodated in the hotel. We can (simultaneously) move the guest currently in room 1 to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from his current room n to room n+1. After this, room 1 is empty and the new guest can be moved into that room. By repeating this procedure, it is possible to make room for any finite number of new guests."

So the paradox states that "It is also possible to accommodate a countably infinite number of new guests: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and, in general, the guest occupying room n to room 2n, and all the odd-numbered rooms (which are countably infinite) will be free for the new guests."

As it looks we have a mathematical paradox here. But i would pose an argument. Infinity is not actually infinite. Infinity by itself has a meaning "Infinity (symbol: ∞) is an abstract concept describing something without any bound and is relevant in a number of fields, predominantly mathematics and physics. In mathematics, "infinity" is often treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as natural or real numbers." But in reality infinity is not infinite, but it has an END. The same applies to the number of Pi. The number of Pi starts with 3,14... and so forth. One would say that this number is "Infinity", but in reality, there will always be a number in the end and a number coming after it n+1. Now we can argue that n+1 is always infinity but the counter-argument would be that there will always be an END (a number). And the argument for the counter-argument is that there will always be a number after the END. Looking from the perspective of the counter-argument (There will always be an END number in infinity), this paradox is false. If an infinite number of rooms will house an infinite number of guests then when a new guest arrives he can't stay in the hotel since infinity has an END, then infinity rooms will house infinity guests. Thus this logic means that if you would take 1 person from room 1 and put him into room 2 and so forth to infinity, the end would be infinity + 1 (excess). Therefore if the new person comes into infinity (which has an END) and all of the people are moved 1 room forth then there would be an excess of 1 therefore this paradox is false. It can only be false if you look from the perspective that INFINITY has an END (end number).

What is your take on this paradox? I find it quite amusing. What are your thoughts?
Apple

Pierre-Normand

But in reality infinity is not infinite, but it has an END. The same applies to the number of Pi. The number of Pi starts with 3,14... and so forth. One would say that this number is "Infinity", but in reality, there will always be a number in the end and a number coming after it 1+n. — Apple

I don't get your argument at all. What make you think that there is "a number in the end"? (Although Pi is both irrational and transcendental, that is irrelevant to your argument about infinity.) The fraction 1/9 (one ninth) is a rational number but it also has an infinite decimal expansion. It can be expressed as 0.11111... where the three dots signify that there isn't any "1" in the decimal expansion that terminates it. And so is it with the set of the natural numbers {1, 2, 3, ...}; there isn't any number N such that N+1 doesn't belong to that set. What motivates you to claim that the ordered sequence of natural numbers has an "END" then?

Michael

If Hilbert's hotel was full then wouldn't everyone already be in a room?

Pierre-Normand

If Hilbert's hotel was full then wouldn't everyone already be in a room? — Michael

Everyone indeed already is in a room both before and after they all are moved to a new room all at once. But after the move (where, e.g. everyone in room n moved to room 2*n), not every room has someone in it. All the odd-numbered room are freed. That's the apparent paradox. But one way to define an infinite set is: a set such that it can be mapped one-to-one to a proper subset of itself.

Since the set of natural numbers is such an infinite set, there are such mappings. For instance the set of even (positive) numbers is a proper subset of the natural numbers and, indeed, the natural numbers can be mapped one-to-one to the set of even numbers. This is what "happens" when all the guest of Hilbert's Hotel (the natural numbers) are mapped to ("moved to") the even numbered rooms, thus freeing an infinite number of odd numbered rooms. Or also, more trivially, when each guest that is currently in room N is moved to room N+1, thus freeing only the first room. This is also a mapping of the set of the rooms that were formerly occupied to a proper subset of itself (i.e. to the set of all the rooms that still are occupied after the move).

(When two sets (that can be finite or infinite) thus have a one-on-one mapping between them, they are said to have the same cardinality. It is one main achievement of Georg Cantor to have shown that there are infinite cardinalities larger that the cardinality of the natural numbers. Such is the case for the cardinality of the real numbers. But there is no set that has "fewer" elements (i.e. a smaller cardinality) than the set of the natural numbers).

Michael

Everyone indeed already is in a room both before and after they all are moved to a new room all at once. But after the move (where, e.g. everyone in room n moved to room 2*n), not every room has someone in it. All the odd-numbered room are freed. That's the apparent paradox. — Pierre-Normand

Ah, yes. Missed the second part of the paradox.

I assume this paradox only arises in the case of actual infinities?

I wonder if that would count as a reductio ad absurdum of actual infinities (despite Hilbert's defence of them)?

Pierre-Normand

I assume this paradox only arises in the case of actual infinities? — Michael

You may be referring to Aristotle's distinction between actual and potential infinity?

Michael

↪Pierre-Normand

Yes. I think. Perhaps I meant to say an actual actual infinity (i.e. an infinity found in nature, something Hilbert does reject), as opposed to a mathematical actual infinity (i.e. a set)?

Hanover

And if Hilbert's hotel were half full, it'd still have an infinite number of guests, which means when it was fully full, it'd have double the infinite number of guests it had when it was half full.

Did we really need a hotel thought experiment to inform us of one of the infinite number of paradoxes associated with infinity?

unenlightened

I used to work in Hilbert's hotel as a night porter, and this sort of thing happened all the time. To move an infinite number of guests took an hour; half an hour for the first one and half the time for each subsequent one. But it was the kitchen staff who had it worst, and eventually the difficulties of producing an infinite number of cooked breakfasts while keeping the food waste finite gave the chef a nervous breakdown and he set fire to the place. As far as I know it is still burning.

TheWillowOfDarkness

↪unenlightened

Luckily there are always more rooms everyone can run too. It all sounds like some sort of Hellscape mystery-adventure platformer.

Pierre-Normand

I wonder if that would count as a reductio ad absurdum of actual infinities (despite Hilbert's defence of them)? — Michael

Since the result merely is counterintuitive, but doesn't generate an actual contradiction, I don't think it militates against the idea of actual infinities as Cantor conceived of them (whatever one may think of infinities realized in nature -- your actual actual infinities). I am unsure how intuitionistic mathematics deals with all of Cantor's results in therms of potential infinities. It is true that the "paradox" of Hilbert's Hotel doesn't seem to arise from the point of view of merely potential infinities (since the "process" of moving the guests to new rooms in order accommodate new guests is never ending). But it still merely is a pseudo-paradox from the point of view of a Platonist mathematics that makes provision for actual infinities (e.g. actually existing sets that have transfinite cardinalities). Just like the idea of relative simultaneity in the theory of special relativity, the idea of a set that can be mapped on a proper subset of itself just is something that our intuition can be reformed to accommodate when prejudice is overcome.

Pierre-Normand

Luckily there are always more rooms everyone can run too. It all sounds like some sort of Hellscape mystery-adventure platformer. — TheWillowOfDarkness

If a new set of guests arrives that represents the real numbers, then, in that case, Hilbert's Hotel won't be able to accommodate them all.

unenlightened

If a new set of guests arrives that represents the real numbers, then, in that case, Hilbert's Hotel won't be able to accommodate them all. — Pierre-Normand

If they form an orderly queue, they can be accommodated; otherwise they will have to go to Cantor's night shelter which has infinite rooms each of infinite capacity on each of it's infinite floors. Breakfast is not provided.

Pierre-Normand

If they form an orderly queue, they can be accommodated; otherwise they will have to go to Cantor's night shelter which has infinite rooms each of infinite capacity on each of it's infinite floors. Breakfast is not provided. — unenlightened

Real numbers, pretty much by definition, can't form an orderly queue. That would mean that they are countable, which they aren't. But it Cantor's night shelter has just two floors, each of which has just two single rooms, each of which has only two sub-rooms, each of which has only two sub-sub-rooms, etc. ad infinitum, then, yes, they can be accommodated.

unenlightened

People that can't form an orderly queue don't deserve breakfast.

shmik

Hey Apple, I think the reason why you come to the conclusion that there must be an 'end' point is that your thinking of infinity as a process. Pi - or 3.14... doesn't occur temporally, its not one digit coming after the other, rather 'pi' represents the whole thing at once. So there is no end or end + 1 point.

jorndoe

:D Isn't Cantor's night shelter just one infinitely large, continuous room? With regularly spaced signs giving directions? Never visited, but fortunately, no matter where you are, you're always still a finite distance from the main exit.

But in reality infinity is not infinite, but it has an END. — Apple

No, there's no last room, only a first room.
Rather, what's finite, is the distance from any given room to the lobby, or to any other given room.
What's infinite is the number of rooms - like a quantity that's not a number.
I don't think the thought experiment can derive a contradiction.

"Plenty of room at the Hotel California", but unlike Hotel California, you can actually leave Hilbert's Hotel.

"Hilbert's Paradox of the Grand Hotel"

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