Could a hotel with an infinite number of rooms be packed?

[SW VandeCarr]

I think the confusion most likely arose when the question (purely mathematical in nature, as you say) was asked in the context of real life entities (those of a hotel and it's rooms.)

If your argument is that it must be discussed only in mathematical contexts such as infinite sets, I don't think a hotel is the best way to illustrate that because it can only be thought of as a physical entity with real-world limitations, and thus the discussion of the nature of infinity as a real world property was inevitable. I'm sure your grade school students would agree.

Are you serious? You're saying that members of PF might be confused by a purely imaginary model and think that it justified launching off into a completely different context? If I say I can slice a pie into an infinite number of slices and ask you how many slices do you want, do you think that's a serious question about the real world? No, it's ploy to get you to think about the concept of infinity and how you must change your way of thinking to grasp the concept of infinite sets. My students do understand that after about a week (and they are not "exceptional" students).

 

[Evolver]

Are you serious? You're saying that members of PF might be confused by a purely imaginary model and think that it justified launching off into a completely different context? If I say I can slice a pie into an infinite number of slices and ask you how many slices do you want, do you think that's a serious question about the real world? No, it's ploy to get you to think about the concept of infinity and how you must change your way of thinking to grasp the concept of infinite sets. My students do understand that after about a week (and they are not "exceptional" students).

Yes I am serious, there is no way to comprehend infinity because it is defined as such. To apply it to real world things (such as infinitely slicing a pie) is irrelevant because it is not testable. You can not infinitely slice a pie. So all you are doing is inciting a philosophical discussion of whether infinity can exist in the real world or not. (as this forum is a testament to).

Perhaps a better way to think of infinity is in terms of information. Say you take a yard stick, and you divide it into inches. You have accrued a certain amount for information about it. If you divide it into it's atomic constituents then you must derive more information form it still, and if you go further and further down you require more and more information... but when all is said and done, it's still only a yard long.

You cannot describe an imaginary concept with use of real world examples. It's counter-intuitive and our brain can never think of a hotel with infinite rooms or a pie with infinite pieces. Infinity is a concept for mathematical equations only.

 

[D H]

Forget about testability and physical reality. This thread is about a concept from pure mathematics, in particular Hilbert's hotel. Google that term (Hilbert's hotel) and you will find a lot written on the topic.

Briefly, to answer the original question, the answer is yes and no. The answer is "yes" in the sense that every room can have an occupant; the hotel has zero empty rooms. How else you define "packed"?

The answer is "no" in the sense that additional occupants can always be accommodated. Assume rooms are numbered 1,2,3 ... -- and assume every room is occupied. Now a prospective tenant arrives at the front desk. How to fit that person in? Simple. Ask the occupants of room 1 to move to room 2, the occupants of room 2 to move to room 3, and so on. Now room 1 is empty. Even an infinite number of occupants can be accommodated. Suppose there are two grand hotels and both are fully occupied due to an infinitely interesting conference. One of the hotels burns down. How to accommodate that infinite influx of tenants? (I'll leave the answer up to you to figure out.)

 

[Evolver]

Forget about testability and physical reality. This thread is about a concept from pure mathematics, in particular Hilbert's hotel. Google that term (Hilbert's hotel) and you will find a lot written on the topic.

Briefly, to answer the original question, the answer is yes and no. The answer is "yes" in the sense that every room can have an occupant; the hotel has zero empty rooms. How else you define "packed"?

The answer is "no" in the sense that additional occupants can always be accommodated. Assume rooms are numbered 1,2,3 ... -- and assume every room is occupied. Now a prospective tenant arrives at the front desk. How to fit that person in? Simple. Ask the occupants of room 1 to move to room 2, the occupants of room 2 to move to room 3, and so on. Now room 1 is empty. Even an infinite number of occupants can be accommodated. Suppose there are two grand hotels and both are fully occupied due to an infinitely interesting conference. One of the hotels burns down. How to accommodate that infinite influx of tenants? (I'll leave the answer up to you to figure out.)

I'm well aware of Hilbert's Paradox. And yes I understand that it is just that... a paradox. One which has also received much criticism for being so incredibly counter-intuitive. If something is not testable, it does not exist as far as science is concerned.

And my point is not that a philosophical/mathematical interpretation of infinity cannot exist, but that one should be wary of using real-life examples to represent an idea that can exist only in mathematical equations. To do otherwise is to incite confusion about the very nature of the subject at all... this forum is a good example of just that.

 

[SW VandeCarr]

You cannot describe an imaginary concept with use of real world examples. It's counter-intuitive and our brain can never think of a hotel with infinite rooms or a pie with infinite pieces. Infinity is a concept for mathematical equations only.

I disagree. What's your evidence for that?

 

[Evolver]

I disagree. What's your evidence for that? This attitude is why so many students become phobic about math. Most of my students (not all), who generally come from economically disadvantaged backgrounds understand this is a model. They are not confused, and they understand that if they had to survive by eating any (finite) number of slices of the pie, they'd starve to death. Many of children I teach go on to high school and can take junior and senior level math courses such as Calc I and II in their freshman and sophomore years. (however, high school Calc I and and II is not as advanced as I and II levels at the college level). They already understand the principles. My coworker volunteer teaches algebra using visual aids. She is also successful at getting her students into advanced classes when they enter magnet high schools. In another one or two years, some of our former our students will be taking their SATs. I'm confident they will do well.

I have no doubt about the abilities of your students. This isn't about that at all. My point is, if your students realize they would starve from eating an infinite amount of pie... yet the you ask them to fill an infinite amount of rooms with an infinite amount of people, or to say the universe may be infinite... have they really grasped what it means for something to be infinite? Or have they just accepted something that was taught to them and which they can reproduce. All infinite has come to mean in this sense, is something which is undefined.

As far as what my evidence is? Infinity cannot be measured, tested or defined by scientific applications. It is not me that must provide evidence... it is you.

 

[SW VandeCarr]

As far as what my evidence is? Infinity cannot be measured, tested or defined by scientific applications. It is not me that must provide evidence... it is you.

I deleted everything but the first sentence of the post you quoted because the unions oppose what we do and it leads off topic.

It seems you still don't get the difference between empirical science and mathematics. Empirical propositions require empirical evidence (they are never proved in the formal sense). Mathematical propositions require mathematical proofs. The evidence I asked for was if the brain indeed can't grasp the difference between an imaginary model based on real world objects and the real world itself. That's an empirical proposition that requires evidence. Since you made the statement, it's up to you to provide evidence.

EDIT: You misread my example in the quoted post. The children eat a finite number of slices from an infinite number of slices.

 

[Evolver]

I deleted everything but the first sentence of the post you quoted because the unions oppose what we do and it leads off topic.

It seems you still don't get the difference between empirical science and mathematics. Empirical propositions require empirical evidence (they are never proved in the formal sense). Mathematical propositions require mathematical proofs. The evidence I asked for was if the brain indeed can't grasp the difference between an imaginary model based on real world objects and the real world itself. That's an empirical proposition that requires evidence. Since you made the statement, it's up to you to provide evidence.

No it is still you that requires the proof, and that is why it is known as Hilbert's Hotel Paradox. There is a yes AND no answer available to the question and as such it is not correctly understood, and it shows that the brain cannot grasp this concept as is currently presented.

EDIT: You misread my example in the quoted post. The children eat a finite number of slices from an infinite number of slices.

I apologize for misreading, but the point remains the same... even if you could slice a pie into an infinite amount of sections, the students would still only have a finite amount of pie to eat at the end of the day. So from this example, it shows that infinity is nothing more than an interpretation of finite qualities. Whether that's true or not, isn't the point, but that's what the example displays and is also a byproduct of associating a highly theoretical concept with real world objects.

 

[debra]

Isnt it a language issue - we are thinking of infinity as a discrete number which it is not. What about: Could a hotel with an infinite number of rooms be packed with an infinite number of guests? Answer - yes?

 

[Evolver]

Isnt it a language issue - we are thinking of infinity as a discrete number which it is not. What about: Could a hotel with an infinite number of rooms be packed with an infinite number of guests? Answer - yes?

Right I believe it more of a one way street and the question itself is too broad. A more specific question (without repercussions) is required. For instance, can a hotel with infinite rooms be filled with and infinite number of guests... yes. Can an hotel with an infinite number of guests in it's infinite number of rooms add an infinite number of new guests... yes.

These can all be said to be true without contradicting the others. The results that these types of questions give aren't necessarily mutually exclusive, just more of a thought experiment.

 

[SW VandeCarr]

No it is still you that requires the proof, and that is why it is known as Hilbert's Hotel Paradox. There is a yes AND no answer available to the question and as such it is not correctly understood.

Why should I show evidence for a statement you made?

Do you know the definition of 'paradox'?

apologize for misreading, but the point remains the same... even if you could slice a pie into an infinite amount of sections, the students would still only have a finite amount of pie to eat at the end of the day.

They will have no pie to eat. If you don't understand the mathematical concept of infinity, how can you argue against it?

EDIT: I just told you the difference between empirical propositions and mathematical propositions. Only the latter requires proof. Your empirical statement cannot be proved but does require evidence. You seem to be unable to grasp the difference between empirical science and formal systems.

 

[debra]

Right I believe it more of a one way street and the question itself is too broad. A more specific question (without repercussions) is required. For instance, can a hotel with infinite rooms be filled with and infinite number of guests... yes. Can an hotel with an infinite number of guests in it's infinite number of rooms add an infinite number of new guests... yes.

These can all be said to be true without contradicting the others. The results that these types of questions give aren't necessarily mutually exclusive, just more of a thought experiment.

We are saying that a hotel with the greatest number possible of rooms is filled by the greatest number possible of guests. So we cannot go on to say - add another greatest number possible to the linguistically already greatest number possible, because that does not make sense then - its already the greatest number possible?

 

[Evolver]

Why should I show evidence for a statement you made?

Do you know the definition of 'paradox'?

Taken from the dictionary:

1. a statement or proposition that seems self-contradictory or absurd but in reality expresses a possible truth.
2. a self-contradictory and false proposition.
3. any person, thing, or situation exhibiting an apparently contradictory nature.
4. an opinion or statement contrary to commonly accepted opinion.

You will note that definitions number 1 and 2 have different meanings... do YOU know what a paradox is?

Besides, you are saying you wish to see if the brain can't comprehend the difference... well then it is you that must prove that result. I am simply finding flaws in your logic.

They will have no pie to eat. If you don't understand the mathematical concept of infinity, how can you argue against it?

No they will still have the full pie... if you cut a pie into 4 pieces, you still have a full pie. If you cut a pie into infinity pieces, you still have a whole pie... just an infinitely undefined number of pieces. It all depends if you are using a countable or uncountable mathematical set for infinity... both of which mathematics utilizes. it is possible for one infinite set to contain more things than another infinite set. But since the Hotel Paradox doesn't define which set is which, there lies the paradox.

 

[Evolver]

We are saying that a hotel with the greatest number possible of rooms is filled by the greatest number possible of guests. So we cannot go on to say - add another greatest number possible to the linguistically already greatest number possible, because that does not make sense then - its already the greatest number possible?

No because the question doesn't define many aspects of the mathematical infinity. Are they countable or uncountable sets? Which set is allowed more than the other. It is an undefined question.

 

[SW VandeCarr]

You cannot describe an imaginary concept with use of real world examples. It's counter-intuitive and our brain can never think of a hotel with infinite rooms or a pie with infinite pieces. Infinity is a concept for mathematical equations only.

This is what you said. What's the evidence that our brain can never think of a hotel with an infinite number of rooms? Conceptually, it's done all the time. The problem can be stated very specifically.

RE your post #48

'Paradox' like may words, has more than one definition. It can be an apparent contradiction which in fact may be true. Because the word is ambiguous, it's not the basis for a yes/no answer.

Take a finite number 'a' from a set with x members such that a<x; then a/x goes to zero at the limit as x goes to infinity.

 

[Hurkyl]

Yes I am serious, there is no way to comprehend infinity because it is defined as such.

Well, maybe that the problem, then. You're interpreting the word "infinity" in some strange way that nobody else is using.

 

[Evolver]

This is what you said. What's the evidence that our brain can never think of a hotel with an infinite number of rooms? Conceptually, it's done all the time. The problem can be stated very specifically.

RE your post #48

'Paradox' like may words, has more than one definition. It can be an apparent contradiction which in fact may be true. Because the word is ambiguous, it's not the basis for a yes/no answer.

Take a finite number 'a' from a set with x members such that a<x; then a/x goes to zero at the limit as x goes to infinity.

Agreed, but there are also multiple definitions of an infinite set. It can be a countable infinite set or an uncountable infinite set. Some infinite sets are allowed to be greater or lesser than other infinite sets. The hotel paradox does not define any of these qualities of an infinite set and thus leave it to interpretation.

The brain cannot fathom an infinite set that is essentially endless. They may be able to use logic to deduce certain implications of the infinite set (like if it's infinite it can always have more rooms, etc.) But that doesn't really mean anything in the essence of truly understanding infinite. Because as I say, it's a placeholder for an unfathomable idea. For instance, to give an analogy from physics... when something becomes 'infinite' it is when the rules of physics break down. (Ex., singularity, infinite mass at light speed, etc.) Scientists don't know what happens in the singularity or even attempt to know, but to say it is infinite is a placeholder for that thought. In mathematics it is more conceptual in nature, but it does not mean it is understood in any classical sense.

 

[Evolver]

Well, maybe that the problem, then. You're interpreting the word "infinity" in some strange way that nobody else is using.

There are multiple definitions of an infinite set... countable, uncountable, greater or lesser, etc... I'm very well aware of the ways in which to interpret it.

 

[Hurkyl]

Agreed, but there are also multiple definitions of an infinite set.

AFAIK, there is pretty much one definition of "infinite set" -- not finite. (With "finite set" meaning, roughly, that its cardinality is a natural number)

What you subsequently describe is that there exists more than one set that is infinite, and that their cardinal numbers may be of different sizes.

The brain cannot fathom an infinite set that is essentially endless.

I am not limited by your lack of imagination. And besides, there are infinite ordered sets that do have ends. $\omega + 1$, for example.

(P.S. it doesn't make sense to talk about the "end" of a set, so I assume you were talking about ordered sets)

And besides, I'm pretty sure I have an easier time "fathoming" set of cardinality $\aleph_0$ than I would "fathoming" a set of cardinality 129848300199285771.

Because as I say, it's a placeholder for an unfathomable idea.

You may be right about whatever you mean by "infinite", but you are the only person talking about that -- the rest of us are content to talk about mathematics and set theory.

For instance, to give an analogy from physics... when something becomes 'infinite' it is when the rules of physics break down. (Ex., singularity, infinite mass at light speed, etc.)

The rules of physics only break down when something becomes infinite if they rules of physics say they do.


Note, for example, that classical mechanics has absolutely no problem with the fact that the number of points within a box has cardinality $2^{\aleph_0}$...


Also note, by the way, that all this talk about infinite sets and cardinality and whatnot has nothing to do with the extended real numbers $+\infty$ and $-\infty$ that arise in calculus and physics.

Scientists don't know what happens in the singularity or even attempt to know,

You're talking about a black hole, right? FYI, in GR, the "singularity" is simply the edge of the universe.

 

[Evolver]

AFAIK, there is pretty much one definition of "infinite set" -- not finite. (With "finite set" meaning, roughly, that its cardinality is a natural number)

What you subsequently describe is that there exists more than one set that is infinite, and that their cardinal numbers may be of different sizes.

True infinite sets are not finite, but that does not mean all infinite sets are the same... Cantor's Diagonal Argument shows that. There are countable and uncountable infinite sets. Some are allowed to be greater than others. For instance integers vs. real numbers... those encompass two separate infinite set types with two different qualities.

I am not limited by your lack of imagination. And besides, there are infinite ordered sets that do have ends. $\omega + 1$, for example.

(P.S. it doesn't make sense to talk about the "end" of a set, so I assume you were talking about ordered sets)

And besides, I'm pretty sure I have an easier time "fathoming" set of cardinality $\aleph_0$ than I would "fathoming" a set of cardinality 129848300199285771.

Imagination has nothing to do with it... it's about having the capacity to comprehend something. A mouse couldn't comprehend calculus no matter how much it tried... would I then say the mouse has no imagination, or simply that it is not adapted to comprehend such things? A mouse would have equal difficulty comprehending advanced calculus as it would high school geometry.

I'm not arguing against humanity or set theory or anything. I'm just calling a spade a spade in the sense that the only way we can attain higher thought is through representative notions and models... and that is precisely what set theory is.

The rules of physics only break down when something becomes infinite if they rules of physics say they do.

But of course, if there were no rules there would be no ideas... set theory has rules too... Like the hotel paradox for example. In the hotel paradox, a hotel with infinite rooms filled by infinite guests can accommodate a new guest, or a bus load of guests, or a bus with infinite new guests, or infinite buses with infinite guests. Infinity modified by 1, or 20, or even infinity is still infinity. Yet there are infinites which can be greater or lesser than other infinites. These too, only exist because set theory says they do. Does that mean you can comprehend it any more or less than you can the laws of physics breaking down?

You're talking about a black hole, right? FYI, in GR, the "singularity" is simply the edge of the universe.

Singularity as in the point where physics (and mathematics) break down, whether that be the crushing gravity of a black hole, or the singularity at the beginning of the big bang. A mathematical singularity is no different... it is the point where a mathematical object is not defined.

 

[disregardthat]

Imagination has nothing to do with it... it's about having the capacity to comprehend something. A mouse couldn't comprehend calculus no matter how much it tried... would I then say the mouse has no imagination, or simply that it is not adapted to comprehend such things? A mouse would have equal difficulty comprehending advanced calculus as it would high school geometry.

Imagination may have something to do with it if certain aspects of infinity is in principle unimaginable.

 

[debra]

The infinity notions have practical problems that even the universe would meet. There must be a limit to the degree of precision in a system or process.

A curve for example must be discontinous at some level, otherwise the steps needed to produce a curve would need to be infinitely small (pixelated). The metric of space time needed to define such small steps would be unmanagabley large.

A computer with a mega trillion bit processor using half the universes energy would still fail to draw a perfect curve.

 

[Pythagorean]

Under what conditions would it be possible for the universe to even be able to define an infinite quantity that we could satisfactorily interpret?

This is exactly why OR rules it out. It's not very helpful to us if we can't access it and make use of it.

 

[Pinu7]

This would depend on what you mean by "infinite amount"
There are two types of infinity

1. Countable infinity("Aleph-null"): A set has the cardinality of countable infinity if a bijective map can be created with the set of natural numbers.

2. Uncountable infinity("continuum" or "c"): A set has the cardinality of infinity if a bijective map
exists between the set and the interval [0,1] on the real number line.
(See: Real Analysis-Kolmogorov)

Denote the set of rooms by R, and the set of guests by A. There are five possibilities(assuming R is infinite).
1. card(R)=Aleph-null, card(A)=/=Aleph-null this means it will not be packed.
2. card(R),card(A)=Aleph-null this will be packed.
3. card(R)=c, card(A) is a natural number there will be infinite(c) free rooms.
4. card(R)=c, card(A)=c this will be packed
5. card(R)=c, card(A)=Aleph-null It will be packed, and there will be unroomed
guests

 

[Tinyboss]

There are infinitely many distinct infinite cardinalities, not just two. Cantor's diagonalization argument assures that the power set of S has cardinality strictly greater than |S|. So |S| < |P(S)| < |P(P(S))| < ...

Edit: I realize that all infinite sets are either countable or uncountable, but the fact that there are distinct uncountable cardinalities means we can't reduce this to a finite number of cases.