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

[Richard87]

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

 

[MaxwellsDemon]

Only if there is an uncountable infinity of guests. Maybe if the Real Numbers decide to hold a convention.

 

[Richard87]

the hotel of interest has infinitely many rooms, so if you move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, you can fit the newcomer into room 1. By repeating this procedure, it is possible to make room for a countably infinite number of new clients: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and in general room N to room 2N, and all the odd-numbered rooms will be free for the new guests.

Apparantly an hotel with an infinite amount of rooms could never be packed, even with an infinite number of guests.

 

[SW VandeCarr]

Apparantly an hotel with an infinite amount of rooms could never be packed, even with an infinite number of guests.

Assuming one guest to a room, there can be a one-to one correspondence between guests and rooms. So the hotel would be "full" but they could always make room for more because \infty+1=\infty

 

[Evolver]

This is a flawed argument because what the concept "infinite" actually equates to in mathematics/physics is "undefined". For example, in physics it is said that in order to make matter attain the speed of light it must have infinite energy... which comes out in equations as undefined. In black holes it is said that gravity compresses matter to such a condensed state that it has infinite density... which comes out to undefined in equations. And that is no less true than the question you have proposed. You have not defined how many rooms must be filled therefore there can be no answer.

 

[debra]

The idea of infinity is fine - a number that never ends, but its practical realization cannot be realized because it would imply an infinite information packet size - which cannot exist in reality. So, following on from that concept, the 1/r equation for increasing field strengths must cut-off at some level. There cannot be infinite field strengths for the same reason.

I find Pi is very interesting in that it appears never to be fully defined. There is a philosophical reason that must be so.

 

[SW VandeCarr]

This is a flawed argument because what the concept "infinite" actually equates to in mathematics/physics is "undefined". For example, in physics it is said that in order to make matter attain the speed of light it must have infinite energy... which comes out in equations as undefined. In black holes it is said that gravity compresses matter to such a condensed state that it has infinite density... which comes out to undefined in equations. And that is no less true than the question you have proposed. You have not defined how many rooms must be filled therefore there can be no answer.

Infinity is in fact a valid concept subject to rigorous treatment in set theory. Even in physics cosmologists are considering the possibility of an infinite universe. The hotel example is an old one often used in teaching students about infinite sets.

In terms of my argument: 'The hotel is full but we can always make room for more.': you might ask if every room has a guest (one to one correspondence) how can there be a vacant room for a new guest? There is an answer. Can you figure it out?

 

[Evolver]

Infinity is in fact a valid concept subject to rigorous treatment in set theory. Even in physics cosmologists are considering the possibility of an infinite universe. The hotel example is an old one often used in teaching students about infinite sets.

In terms of my argument: 'The hotel is full but we can always make room for more.': you might ask if every room has a guest (one to one correspondence) how can there be a vacant room for a new guest? There is an answer. Can you figure it out?

Infinity is still undefined by nature though. If something were truly infinite it would have no reason to define itself because doing so would impose a physical limitation. When scientists use infinity as a concept they do so in the sense that they also use virtual particles or imaginary numbers. There is no physical version of them, they simply represent a concept that cannot be otherwise described. Hawking's black body radiation uses virtual pairs at the event horizon of a black hole... it is a concept to explain that phenomenon.

 

[Hurkyl]

When scientists use infinity as a concept they do so in the sense that they also use virtual particles or imaginary numbers.

Or, y'know, integers. The natural number 1 is no less abstract than anything else you listed there -- its only distinction that it, and some applications, were taught to you at a very young age.

 

[Evolver]

Or, y'know, integers. The natural number 1 is no less abstract than anything else you listed there -- its only distinction that it, and some applications, were taught to you at a very young age.

Very true, that is a good point, Hurkyl. It makes my point about infinity being undefined even more true, because 1 is essentially no different than infinity to the universe, these are all man made concepts and as such only produce man made results. So the whole hotel question is already biased based on the idea that the universe would never produce such a paradox, only man can abstract this concept from exaggerated observations of the universe itself.

 

[SW VandeCarr]

Very true, that is a good point, Hurkyl. It makes my point about infinity being undefined even more true, because 1 is essentially no different than infinity to the universe, these are all man made concepts and as such only produce man made results. So the whole hotel question is already biased based on the idea that the universe would never produce such a paradox, only man can abstract this concept from exaggerated observations of the universe itself.

Present observations seem to indicate the universe is flat which raises the real possibility that it is Euclidean and infinite. I started a topic recently in the Cosmology Forum entitled "Questionable article in Scientific American?" and linked to a summary of an article by physicist Max Tegmark regarding an infinite universe. It is so outlandish that it sounded frankly cranky to me. I was informed that though it is extremely weird it is not science fiction, but reasonable scientific speculation and not in violation of the laws of physics as they are currently known. The principle objections to an infinite universe are philosophical, not scientific.

 

[Evolver]

Present observations seem to indicate the universe is flat which raises the real possibility that it is Euclidean and infinite. I started a topic recently in the Cosmology Forum entitled "Questionable article in Scientific American" and linked to a summary of an article by physicist Max Tegmark regarding an infinite universe. It so outlandish that sounded frankly cranky to me. I was informed that though it is extremely weird it is not science fiction, but reasonable scientific speculation and not in violation of the laws of physics as they are currently known.

Well the problem with speculation is that it completely rides on what theory you subscribe to. According to Relativity, the universe is constantly stretched and distorted. According to String Theory it has upwards of 9 spatial dimensions and a time dimension. M-theory indicates that it is the result of two interacting, higher dimensional branes.

Once you start dealing with alternate dimensions, ideas of infinity in a finite space arise, much like the Klein Bottle. At our current limited state of understanding, it is nothing more than speculation, though I do think it is an interesting idea.

 

[SW VandeCarr]

Well the problem with speculation is that it completely rides on what theory you subscribe to. According to Relativity, the universe is constantly stretched and distorted. According to String Theory it has upwards of 9 spatial dimensions and a time dimension. M-theory indicates that it is the result of two interacting, higher dimensional branes.

Once you start dealing with alternate dimensions, ideas of infinity in a finite space arise, much like the Klein Bottle. At our current limited state of understanding, it is nothing more than speculation, though I do think it is an interesting idea.

Very fast response. I just wanted you to know I added one more sentence just after first posting: The principle objections (to an infinite universe) are philosophical, not scientific.

 

[Evolver]

Very fast response. I just wanted you to know I added one more sentence just after first posting: The principle objections (to an infinite universe) are philosophical, not scientific.

I totally agree, and personally I'm not against the idea. I just know that, to this date that we are aware of, the universe has never defined anything as being truly infinite (because by definition if you defined it it could not be infinite.)

I see infinite as more of a man-made concept as opposed to a property of the universe... though I could be completely wrong.

 

[DocZaius]

Very fast response. I just wanted you to know I added one more sentence just after first posting: The principle objections (to an infinite universe) are philosophical, not scientific.

Could you elaborate on what types of philosophical objections there would be to an infinite universe, aside from "I don't like it that way?" Not saying you have such objections, just curious about what's being said.

 

[Evolver]

Could you elaborate on what types of philosophical objections there would be to an infinite universe, aside from "I don't like it that way?" Not saying you have such objections, just curious about what's being said.

Yes. The universe has never defined any infinite quality of itself in any way that we are yet aware of. So in actuality, assuming the universe could be infinite is the actual use of philosophy, not the other way around.

 

[DocZaius]

Yes. The universe has never defined any infinite quality of itself in any way that we are yet aware of. So in actuality, assuming the universe could be infinite is the actual use of philosophy, not the other way around.

I think assuming either way is philosophizing. On matters untestable by science, isn't it best to say "I don't know?" That is why I asked my original question. I can't think of any philosophical objection (on either side of the issue) that would hold any weight.

 

[SW VandeCarr]

Could you elaborate on what types of philosophical objections there would be to an infinite universe, aside from "I don't like it that way?" Not saying you have such objections, just curious about what's being said.

One of the consequences of an infinite universe is that's its populated by an infinite number of people just like us. Indeed, they can be identical copies of us. Each one of us has an infinite number of copies. What does that say about the autonomy of the individual? If you are an identical copy of another individual, not just a biological twin, but in every detail of your life including your history. memories parents, ect; then why isn't that person you? How are you distinguished from the other person? Does everyone's life repeat over and over again in some distant place?

If you think what I just said is whacko, I agree, but it is a consequence of infinity. In an infinite universe anything that can happen will happen, and will happen over and over again an infinite number of times!

Not being a philosopher. I don't know if this a valid objection, but I know that significant parts of philosophy as a discipline are devoted to the nature of the individual and the nature of existence.

 

[Evolver]

I think assuming either way is philosophizing. On matters untestable by science, isn't it best to say "I don't know?" That is why I asked my original question. I can't think of any philosophical objection (on either side of the issue) that would hold any weight.

In matter's unsure to science usually Occam's Razor is invoked, saying that if faced with many possibilities that lead to the same answer that the least complex should be taken until proven otherwise. It prevents unnecessary assumptions and cuts out non-important factors.

In this case, saying that the universe has never defined any infinite quantity thus far, so it's safe to assume it's not infinite, would be invoking Occam's Razor. Speculating that it's infinite (though always possible) is philosophy at this current time.

 

[DocZaius]

In matter's unsure to science usually Occam's Razor is invoked, saying that if faced with many possibilities that lead to the same answer that the least complex should be taken until proven otherwise. It prevents unnecessary assumptions and cuts out non-important factors.

In this case, saying that the universe has never defined any infinite quantity thus far, so it's safe to assume it's not infinite, would be invoking Occam's Razor. Speculating that it's infinite (though always possible) is philosophy at this current time.

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

From my viewpoint it seems that Occam's Razor would imply an interpretation of infinity, at the very least on the micro scale. There is no reason to believe that the fabric on the microscale isn't continuous. And if your objection is Planck length, my understanding (although it could be wrong) of Planck length is that despite its use in physics, it does not physically signify a "smallest possible distance"; which would imply that the universe is made of discrete parts.

My observation of what seems to be continuity everywhere (and that is enough for OR) suggests infinitesmall intervals and I invoke Occam's Razor in that sense.

My point being that I don't think OR favors either interpretation.

 

[Evolver]

Under what conditions would it be possible for the universe to even be able to define an infinite quantity?

From my viewpoint it seems that Occam's Razor would invoke an interpretation of infinity, at the very least on the micro scale. There is no reason to believe that the fabric on the microscale isn't continuous. And if your objection is Planck length, my understanding (although it could be wrong) of Planck length is that despite its use in physics, it does not physically signify a "smallest possible distance"; which would imply that the universe is made of discrete parts.

Well that's my point. Infinity is a concept created by man to represent a thought he cannot comprehend. But it doesn't mean that it is a truth of the universe. The universe doesn't care about infinity as much as man does ;) I think it's possible we are thinking about it in the wrong way.

Here is another post of mine that deals with the universe as something completely different than what we intuitively think: https://www.physicsforums.com/showthread.php?t=368284

 

[debra]

Infinity is in fact a valid concept subject to rigorous treatment in set theory.

A rigorous treatment in set theory? I am not impressed. Should I be over-awed by that statement?

 

[Hurkyl]

I find Pi is very interesting in that it appears never to be fully defined. There is a philosophical reason that must be so.

Ack, I can't believe I missed this. What could you possibly mean by that assertion? Or are you doing something weird like confusing "well-defined number" with "number whose decimal expansion has finitely many nonzero digits"?

 

[D H]

This is a flawed argument because what the concept "infinite" actually equates to in mathematics/physics is "undefined".

The concept of infinity is quite well-defined.

I find Pi is very interesting in that it appears never to be fully defined. There is a philosophical reason that must be so.

Pi is also very well-defined. For example, see http://en.wikipedia.org/wiki/List_of_formulas_involving_π. Just because there remain some open questions about pi (e.g., whether pi is normal) does not mean pi is not well-defined.

 

[debra]

There is no reason to believe that the fabric on the microscale isn't continuous. And if your objection is Planck length, my understanding (although it could be wrong) of Planck length is that despite its use in physics, it does not physically signify a "smallest possible distance"; which would imply that the universe is made of discrete parts.

Then what do you do with the force is proportional to 1/r problem? Implying infinite forces.
Infinities problems dog physics - they have no answer. Or, let's introduce a cut-off because that is what experiment shows. To define small separations needs massive data chunks.
These data chunks - called metrics in physics - would need to go to infinity if space were continous. Sorry - space must be discrete at the micro level.

 

[Evolver]

The concept of infinity is quite well-defined.Pi is also very well-defined. For example, see http://en.wikipedia.org/wiki/List_of_formulas_involving_π. Just because there remain some open questions about pi (e.g., whether pi is normal) does not mean pi is not well-defined.

Infinity is only well defined in that is is a way to represent an idea that cannot be comprehended. It has no other purpose than a place holder for that abstract thought. The universe itself has given no concrete example of any infinite part of itself other than as an abstract placeholder in mathematical equations.

 

[debra]

Ack, I can't believe I missed this. What could you possibly mean by that assertion? Or are you doing something weird like confusing "well-defined number" with "number whose decimal expansion has finitely many nonzero digits"?

OK my exact math definitions may be lacking. I do information science nowadays. But I meant "number whose decimal expansion has finitely many nonzero digits"
I assume Pi has an infinite number of non-zero digits to be philosophically consistent - in a sense.

 

[debra]

What I am trying to say is that a curve cannot be a true representation where space is discrete can it? At a micro micro level there is no 'curve' at all. Think of it as pixelated if you want an easy mind picture. So there can in reality be no such entity as a curve. It can only be curved to so many degrees of precision. Again, at heart, its an infinity problem yet again isn't it? Is space continuous - I say no! There is a philosophical problem with a so-called curve - it is better thought of as a summation of steps.

 

[DocZaius]

What I am trying to say is that a curve cannot be a true representation where space is discrete can it? At a micro micro level there is no 'curve' at all. Think of it as pixelated if you want an easy mind picture. So there can in reality be no such entity as a curve. It can only be curved to so many degrees of precision. Again, at heart, its an infinity problem yet again isn't it? Is space continuous - I say no! There is a philosophical problem with a so-called curve - it is better thought of as a summation of steps.

So your argument is that for spacetime to be continuous, that would mean that infinity would be involved. And therefore it can't be. Why can't infinity be involved at scales we cannot measure? Why is that a logical impossibility? As far as I know the quantization of spacetime is still a postulate and is actively researched. Also, the calculus summation of infinitesmall intervals is a mathematical tool. It does not prove or disprove the physicality of a discrete spacetime.

 

[SW VandeCarr]

A rigorous treatment in set theory? I am not impressed. Should I be over-awed by that statement?

What's your point? Who cares if you're impressed or not?'

 

[WaveJumper]

This discussion is going in the wrong direction. It's not that there is no infinity in Nature, quite the opposite. The question is why we see discrete objects that we label with integers?

Everything in nature is in fact infinite, the wavefunction of every single object in the universe is soaked in around it to infinity(consequence of SE; you could also think of it as it's represented in QFT - fields vs partilces). The real question is why we see single laptops, single cats, single everything, which is the heart of the measurement problem. We are inferring that "objects" are not infinite based on how we perceive them, not how they truly are, apart from our perception of them. Our perception is incredibly flawed and at odds with most everything that has come out of physics in the last 100 years.
The only way around 'infinity-discreteness' is to solve the MP. The solution may turn everything we have assumed so far upside down.

 

[debra]

So your argument is that for spacetime to be continuous, that would mean that infinity would be involved. And therefore it can't be. Why can't infinity be involved at scales we cannot measure? Why is that a logical impossibility? As far as I know the quantization of spacetime is still a postulate and is actively researched. Also, the calculus summation of infinitesmall intervals is a mathematical tool. It does not prove or disprove the physicality of a discrete spacetime.

A metric defines space-time in terms of x,y,z,t. Thus a particle (or field) knows its position in space-time or, simply stated, its distance away form another particle with which it interacts.
For space to be continuous the metric that defines it would have to be infinite precision. There must be a limit to the amount of data an interaction can process. That limit is a constant of the universe - in QFT we give it a cut-off distance. It must 'know' the separation and then act according to its algorithm for that interaction. So the algorithm must be using the separation metrics and as the separation becomes close to zero, the amount of data in that metric value becomes infinite. Of course, it cannot become infinite, so there is a cut-off or a discrete nature of space itself.

 

[debra]

This discussion is going in the wrong direction. It's not that there is no infinity in Nature, quite the opposite. The question is why we see discrete objects that we label with integers?

Everything in nature is in fact infinite, the wavefunction of every single object in the universe is soaked in around it to infinity(consequence of SE). The real question is why we see single laptops, single cats, single everything, which is the heart of the measurement problem. We are inferring that "objects" are not infinite based on how we perceive them, not how they truly are, apart from our perception of them. Our perception is incredibly flawed and at odds with most everything that has come out of physics in the last 100 years.
The only way around 'infinity-discreteness' is to solve the MP. The solution may turn everything we have assumed so far upside down.

Interesting point. But the discreteness of large objects such as a laptop is an illusion. If the laptop were a single particle, it would only give a value when it was observed by a photon or other particle. If it is not being observed then it essentially does not exist in space-time. It only has a probability of being somewhere in its wave function envelope. What we think of as actually 'there' is an observation value. A quantum prepared particle going from A to B is described by a wave function. There is no record of the path it took from A to B, (it was the wave function, not the particle travelling) there are only values it gave when observed at A and then at B. Essentially the particle was a wave function in between and not actually 'existing' like we think a laptop exists. The wave function tells us where it would possibly have been if we had observed it.

The laptop is an illusion because there are mega-trillions of wave functions confined to, essentially, the volume of the laptop. They are firing off observations by the trillion but each is an observation of where a particle is when it is observed. The reality is that each particle is somewhere within its wave function envelope that is the same as one particle going from A to B as previously mentioned. The fact that we think the laptop is a discrete object is an illusion.Its a collection of observations. Not a single part of the laptop exists unless it is 'observed' by an interaction with another particle or external photon.

So the laptop is really observing end points of particles going from A to B (or confined in a stationary Heisenberg area). If we are not observing a particle then there is no location for it. And an observation is simply a value returned at one particular instant to an enquirer.

 

[SW VandeCarr]

This discussion is going in the wrong direction.

It certainly is. To begin with, the OP was a about an imaginary hotel: just in case some people didn't understand that. The discussion of real physical infinity is off topic and in any case, not an issue that can be definitively addressed by current science in terms of whether spacetime is infinite at the smallest or largest scales.

The subject was infinite sets, a purely mathematical concept. If you want to argue that a mathematical argument is flawed, then you have to argue from a mathematical perspective. Even grade school children that I teach as a volunteer understand that.

Perhaps some people thought an infinite hotel was real. Well if was, it could blow out the competition by charging really low prices.

 

[Evolver]

It certainly is. To begin with, the OP was a about an imaginary hotel: just in case some people didn't understand that. The discussion of real physical infinity is off topic and in any case, not an issue that can be addressed by current science.

The subject was infinite sets, a purely mathematical concept. If you want to argue that a mathematical argument is flawed, then you have to argue in terms of a mathematical context. Even grade school children that I teach as a volunteer understand that.

Perhaps some people thought an infinite hotel was real. Well if was, it could blow out the competition by charging really low prices.

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.

 

[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.