What is the issue with an infinite universe on a N-sphere?

[StJohnRiver]

There are physicists who insist that the universe is finite and has a distinct geometry. So what'd be the problem if the universe were infinite?

 

[Drakkith]

Those physicists need to look into modern cosmology then. It is not known whether the universe is finite or infinite, and it doesn't matter, according to current theory, which one is correct. If the universe is finite then it must be much larger than the observable universe. If it is infinite, well, then it is infinite. As for the geometry of the universe, it has been observed to be very very close to being flat, but this does not rule out non-flat geometry that is too subtle to detect currently.

 

[Pianoasis]

Well due to Zeno's paradox of motion depicted by achilles and the tortoise, infinity does not exist. Zeno proved that space is made up of an ultimately small piece that can not be divided any smaller.
Infinity is a number that cannot be divided, cannot be measured, and cannot be contained. This infinite universe obviously does not exist due to the fact that all pieces of space are made of this ultimately small unit.
Every quantity can be described by this unit, thus making the concept of infinity null.

 

[Matterwave]

Well due to Zeno's paradox of motion depicted by achilles and the tortoise, infinity does not exist. Zeno proved that space is made up of an ultimately small piece that can not be divided any smaller.
Infinity is a number that cannot be divided, cannot be measured, and cannot be contained. This infinite universe obviously does not exist due to the fact that all pieces of space are made of this ultimately small unit.
Every quantity can be described by this unit, thus making the concept of infinity null.

Zeno's paradox has been sufficiently done away with via the use of calculus. In calculus, no such quantization is necessary.

There has been no successful quantization of space-time so far.

 

[San K]

There are physicists who insist that the universe is finite and has a distinct geometry. So what'd be the problem if the universe were infinite?

Infinity as a concept in the mind is ok
but
when applied to/in reality, in my understanding/opinion, does not work.

The definition itself is self contradictory/limiting.

however its possible that if you were to reach the "edge" of say, time-space or any other variable/dimension, you could extend it further but it would still remain finite.

I feel that QM might actually be solving the problem of infinity...at least for some concepts such as the infinite loop of a cause having a cause.

 

[audioloop]

There are physicists who insist that the universe is finite and has a distinct geometry. So what'd be the problem if the universe were infinite?

there is a physical problem, in an infinite universe it have an infinite mass and hence infinite inertia, no motion would be possible.

think rather in a finite universe without boundaries.

 

[my_wan]

Well due to Zeno's paradox of motion depicted by achilles and the tortoise, infinity does not exist. Zeno proved that space is made up of an ultimately small piece that can not be divided any smaller.
Infinity is a number that cannot be divided, cannot be measured, and cannot be contained. This infinite universe obviously does not exist due to the fact that all pieces of space are made of this ultimately small unit.
Every quantity can be described by this unit, thus making the concept of infinity null.

No. What Zeno never mentions as his intervals are halved it the time taken for each step. Hence, with the development of calculus it became easy to prove that even if the absolute number of steps are infinite it is easily accomplished in a finite amount of time.

Some more interesting paradoxes have been proposed since Zeno. Including the "Ant on a rubber rope" and "Hilbert's paradox of the Grand Hotel". Vitali sets are cool and easier to understand than Banach–Tarski paradox. Also the Hausdorff paradox is interesting. Basically trying to avoid infinities is every bit as paradoxical as infinities themselves.

The reason calculus is traditionally defined in terms of limits was to avoid infinities, because people inappropriately assumed too much, like invalid Zeno stuff. It was thus better to avoid it than to throw around Zeno stuff a priori. Once that was done, and proofs about the properties of infinities could be proven anyway, the need to avoid infinities lost its relevance. Hence non-standard calculus was born. If you want to make sense of it you'll need to drop the Zeno stuff and stick with more rigorous justifications, such as provided by non-standard calculus.

 

[my_wan]

This being a physics forum it's also interesting to point out some consequences. If everything must be quantized to avoid infinities then General relativity must be quantized. If everything must be quantized then this is tantamount to the claim that Quantum computers are not fully scalable, i.e., have fundamental limits. Of course you can offload that onto other Universes, but then you are stuck with an infinite number of finite Universes.

That's some fairly serious empirically accessible claims, because it still only avoids infinities by making Quantum computers unscalable.

 

[The_Duck]

there is a physical problem, in an infinite universe it have an infinite mass and hence infinite inertia, no motion would be possible.

No. That's not how inertia works.

 

[Drakkith]

there is a physical problem, in an infinite universe it have an infinite mass and hence infinite inertia, no motion would be possible.

think rather in a finite universe without boundaries.

Perhaps if you think of the entire universe as a system moving relative to another system. Per the usual definition of the universe, that it is everything that exists, this is not possible.

 

[TrickyDicky]

This being a physics forum it's also interesting to point out some consequences. If everything must be quantized to avoid infinities then General relativity must be quantized.

I don't follow you. How did you go from the fact that Calculus solves Zeno paradox (wich it does, at least partially) to the claim that everything must be quantized?
I don't see the connection between Calculus and quantization in physics or in general. Precisely Calculus allows us to deal with infinities in a continuous form.

 

[my_wan]

I don't follow you. How did you go from the fact that Calculus solves Zeno paradox (wich it does, at least partially) to the claim that everything must be quantized?
I don't see the connection between Calculus and quantization in physics or in general. Precisely Calculus allows us to deal with infinities in a continuous form.

I don't. I was reacting to what Pianoasis stated the claim was made that: "Zeno proved that space is made up of an ultimately small piece that can not be divided any smaller."

Even so, my words did not constitute a "claim" as it was prequalified with an "if". Certainly I did go a step beyond Pianoasis's words when I associated their notion of units of space which cannot be subdivided with quantization, though Quanta are not things in the usual sense. But it was a trivial extension which laid the groundwork for what I was rejecting, rather than claiming.

 

[TrickyDicky]

I don't. I was reacting to what Pianoasis stated the claim was made that: "Zeno proved that space is made up of an ultimately small piece that can not be divided any smaller."

Even so, my words did not constitute a "claim" as it was prequalified with an "if". Certainly I did go a step beyond Pianoasis's words when I associated their notion of units of space which cannot be subdivided with quantization, though Quanta are not things in the usual sense. But it was a trivial extension which laid the groundwork for what I was rejecting, rather than claiming.

Ah, I see.

 

[Whovian]

Infinity as a concept in the mind is ok
but
when applied to/in reality, in my understanding/opinion, does not work.

Oh?

The definition itself is self contradictory/limiting.

Which definition? There are a lot of definitions of infinity.

however its possible that if you were to reach the "edge" of say, time-space or any other variable/dimension, you could extend it further but it would still remain finite.

Actually, in most cases, infinite means unbounded, so there would be no such edge.

 

[Naty1]

If the universe were infinte, we'd likely not know it. The Earth would most likely remain in orbit about the sun, and our galaxy would be indistinguishable from the one we observe...
in fact all observations would be the same.

Flatness or near flatness that is usually assumed is relevant to the observable universe. Our inflationary cosmology says nothing of the global geometry of the universe...yet.

 

[bahamagreen]

Some more interesting paradoxes have been proposed since Zeno. Including the "Ant on a rubber rope" and "Hilbert's paradox of the Grand Hotel".

Thanks for mentioning those, I had never run across the Ant On A Rubber Rope before.
Might Hilbert's Hotel have a flaw in the premise that relates to problems with infinity? If an infinite number of rooms are each occupied by a guest, where does the new guest come from? Some think that an infinite collection must necessarily contain all instances...

 

[nikkoo]

Actually, infinite means unbounded, so there would be no such edge

 

[my_wan]

Thanks for mentioning those, I had never run across the Ant On A Rubber Rope before.
Might Hilbert's Hotel have a flaw in the premise that relates to problems with infinity? If an infinite number of rooms are each occupied by a guest, where does the new guest come from? Some think that an infinite collection must necessarily contain all instances...

It was Georg Cantor that demonstrated that not only was actual infinities perfectly logical but that it necessarily entailed orders of infinity, called aleph numbers \aleph or cardinality (a powerset). Cardinality is basically the size of an infinite set. This was controversial in Cantor's day since the only infinity acceptable prior to that was unbounded sets, or potential infinities.

You can look up his work, including his diagonal argument, for a more quantitative treatment. I'll just give a more intuitive description. If you have a finite interval, i.e., distance between two points, then logically you can divide it into an infinite set of infinitesimal points. To answer the above question one way, it cannot be said that the infinite set of points between 0 and 1 contain all possible numbers. There is not only more than one infinite set, there are an infinite set of infinite sets. There is no flaw in Hilbert's Hotel.

There is also countably infinite sets and uncountable sets. There is also open sets and closed sets, which differ only in whether they contain the infinitesimal boundary points or not. There are also dense sets, which play a role in many quantum foundation arguments, including EPR and the scalability of quantum computing.

The one thing you cannot do is make a priori generalized statements, like infinity must contain the entirety of the whole Universe to be infinite. Any finite subset of the Universe also contains an infinite set of infinitesimal points. Neither can you, for the same reason, make the claim that two infinite sets must be the same size. You must restrict your statements about infinity to those statements that can be mathematically demonstrated to be consistent, and avoid the intuitively implied and inconsistent properties Zeno's arguments depended on.

 

[bahamagreen]

To answer the above question one way, it cannot be said that the infinite set of points between 0 and 1 contain all possible numbers. There is not only more than one infinite set, there are an infinite set of infinite sets. There is no flaw in Hilbert's Hotel.

I'm not thinking that the infinite set of points between 0 and 1 contains all possible numbers, only that it contains all possible numbers between 0 and 1. It seems to me by definition, the set of points between 0 and 1 must include every point between 0 and 1. Are you suggesting otherwise?

I'm thinking that any arbitrary number I specify between 0 and 1 must already be included in the set of points between 0 and 1; so I don't see how any possible number between 0 and 1 is not already a member of the set of points between 0 and 1.

The "new guest" coming to Hilbert's Hotel's is like a point between 0 and 1 that is not a member of the set of points between 0 and 1... I see this as a flaw in the premise.

If the 0 to 1 range is problematic, we can do the same with the set of natural numbers... I'm thinking that the set of natural numbers must include any and all arbitrary natural numbers that I may specify... this seems clear by definition.
If each occupied room is mapped to a natural number, an infinite number of rooms means all the natural numbers are mapped, as are their corresponding guests... the "new guest" would need to represent an unmapped natural number, but there are none, by definition.

Maybe I'm missing something...?

 

[Whovian]

The "new guest" coming to Hilbert's Hotel's is like a point between 0 and 1 that is not a member of the set of points between 0 and 1... I see this as a flaw in the premise.

How does it remotely resemble this?

EDIT: Upon closer inspection, the paradox is simply mapping natural numbers to new natural numbers. The set $\mathbb{Z}\cup\left[1,\infty\right)$ has the same cardinality as the set $\mathbb{Z}\cup\left[2,\infty\right)$. This should solve the paradox quite easily.

 

[audioloop]

Actually, in most cases, infinite means unbounded, so there would be no such edge.

or finite without boundaries, like a torus.

 

[my_wan]

I'm not thinking that the infinite set of points between 0 and 1 contains all possible numbers, only that it contains all possible numbers between 0 and 1. It seems to me by definition, the set of points between 0 and 1 must include every point between 0 and 1. Are you suggesting otherwise?

In the original post I addressed it was implied that maybe if there was an infinite number of hotel rooms, then these rooms being occupied implied infinite guest such that there could be no new guest. If there are an infinite set of point between the two points, [0,1], and each of these correspond to a hotel room occupied by a point, then the original suggestion to work around the hotel hotel paradox implies that this infinity of points, [0,1], contains all points that might occupy the infinity of hotel rooms. Hence I made the suggestion in order to provide proof by contradiction that the hotel paradox was not flawed.

I'm thinking that any arbitrary number I specify between 0 and 1 must already be included in the set of points between 0 and 1; so I don't see how any possible number between 0 and 1 is not already a member of the set of points between 0 and 1.

The "new guest" coming to Hilbert's Hotel's is like a point between 0 and 1 that is not a member of the set of points between 0 and 1... I see this as a flaw in the premise.

If the points between 0 and 1 are an infinite set of occupied hotel rooms, and yet "new guest" are still available from members that are not member of the set of points between 0 and 1, why is this a special case? The original suggestion was that an infinite number of guest implied no more guest exist, but here you add a special case to say there are more guest available from sets other that [0,1].

If the 0 to 1 range is problematic, we can do the same with the set of natural numbers...

Precisely. The infinity problem is just as big in the interval [0,1] as it is in the interval [0,∞].

I'm thinking that the set of natural numbers must include any and all arbitrary natural numbers that I may specify... this seems clear by definition.
If each occupied room is mapped to a natural number, an infinite number of rooms means all the natural numbers are mapped, as are their corresponding guests... the "new guest" would need to represent an unmapped natural number, but there are none, by definition. Maybe I'm missing something...?

Only problem is that I can pull new guest from the infinite set of real number which you didn't included here. Note that the numbers are merely name tags on the guest, and it make no difference which ones you label with which numbers. I can relabel an infinite number of guest labeled with even numbers with odd numbers, and visa versa, and the count remains the same. I can also relabel all natural numbers as real numbers simply by multiplying their name tags with an irrational number and assigning them that number. It changes nothing about the total number of guest.

 

[Endervhar]

Cantor showed that all these infinities existed, but we should not lose sight of the fact that they are mathematical infinities. Mathematically, what does it mean to say that something exists? If a mathematician can write down a set of non-contradictory axioms, and set down rules for deducing mathematically true statements from them, then those statements can be said to ‘exist’. This existence requires only logical self-consistency. Physical existence is completely unnecessary.

If there can be a profound difference between physical and mathematical “existence” then it seems reasonable to identify a similar difference between physical and mathematical “truth”. Cantor’s infinities were all mathematical infinities, as are the rooms and guests in Hilbert’s Hotel. They may bear no relation to any possible physical infinity, which would include an infinite universe.

The actual, ancient, fear of infinity was not removed; it was just that Cantor provided the world with a “label” that could be attached to infinity, which reads: “this is a mathematical infinity – it doesn’t bite”.

 

[bahamagreen]

my_wan,
Your responses support my thinking that the Hilbert Hotel premise is flawed.

A "complete" mapping to the natural numbers would be to include all of them by taking the natural numbers in order... 1,2,3... any other mapping scheme like 2,4,6... is an obvious mechanism to skip some numbers, yet there would be objection to a scheme that skipped points between 0 and 1.
Claiming that the new guest could be from the set of real numbers when the set of guests is represented by the natural numbers misses the whole point. If the points between 0 and 1 represent the rooms, it is the guests that are mapped to the natural numbers. The paradox is based on the assumption that all guests, including any possible new guests, are all of the same class of thing - natural numbers, and violating that by suggesting a new guest could be a real number is like solving the question of the origin of the new guest by finding a chair and checking that chair into the hotel as a new guest... no, the new guest has to be a person like all the other guests.

 

[my_wan]

Claiming that the new guest could be from the set of real numbers when the set of guests is represented by the natural numbers misses the whole point.

It doesn't miss the point any more than saying that the set of all hotel customers must consist of all possible hotel customers, and that is the only way you can claim there is nobody remaining to request a room in the hotel.

If the real numbers represent the set of all present hotel customers, and the natural numbers represent the set of all people that might request a room, then there are an infinity of people that may request a room even after an infinite number of people have already filled the hotel. To assume otherwise is effectively an attempt to impose a boundary condition on an unbounded variable.

 

[my_wan]

Cantor showed that all these infinities existed, but we should not lose sight of the fact that they are mathematical infinities.

Very true, and so far the justification for "actual infinities" is fairly slim. However, attempting to avoid them has a number of problems. Avoiding actual infinities is just as problematic and paradoxical as accepting them. Modern mathematics didn't select the axioms simply to avoid contradictions with infinities, they where selected to avoid mathematical contradiction as a result of attempting to avoid them.

In the physical sciences the scalability of quantum computers actually depends on these mathematical properties associated with mathematical infinities. This comes from the fact that a Hilbert space must be "complete", i.e., is a complete metric space. The very thing that allows the calculus of limits, and from which we derive our mathematical justifications for infinities.

 

[ObsessiveMathsFreak]

There are physicists who insist that the universe is finite and has a distinct geometry. So what'd be the problem if the universe were infinite?

It depends on what you mean by infinite. You really have to think about this one.

 

[bahamagreen]

It doesn't miss the point any more than saying that the set of all hotel customers must consist of all possible hotel customers, and that is the only way you can claim there is nobody remaining to request a room in the hotel.

If the real numbers represent the set of all present hotel customers, and the natural numbers represent the set of all people that might request a room, then there are an infinity of people that may request a room even after an infinite number of people have already filled the hotel. To assume otherwise is effectively an attempt to impose a boundary condition on an unbounded variable.

The set of natural numbers must include all possible natural numbers.
The set of points between 0 and 1 must include all possible points between o and 1.
The set of all hotel customers must comprise all possible hotel customers.

You are defining:
The reals is the set of present hotel customers.
The naturals is the set of people that might request a room (not presently hotel customers).

I'm suggesting that is a flaw because you are defining some persons as both a non-customer and a customer - because the naturals and reals share some members in common (all naturals are members of the reals, some reals are members of the naturals).

But maybe I'm still missing something?

 

[my_wan]

The set of all hotel customers must comprise all possible hotel customers.

This can't be justified. The set of all actual hotel customers anywhere in the world does not comprise all possible hotel customers. If that was necessarily true then in order for hotels to have any customers they must have every person on Earth as a customer. Conversely, by this logic, since I am not a customer, either hotels have no customers or I am not a potential customer. Neither of which is true.

 

[Endervhar]

This can't be justified. The set of all actual hotel customers anywhere in the world does not comprise all possible hotel customers.

As ObsessiveMathsFreak pointed out, this "depends on what you mean by infinite." In terms of mathematical infinities your assertion is undoubtedly true, but it is a mathematical "truth" and has no significance in reality. You cannot have an infinite number of rooms, an infinite number of people, or an infinite number of anything.

Earlier, someone suggested that infinite and boundless might be synonymous; this is not so. Anything that is infinite is boundless, but not everything that is boundless is infinite.

 

[Endervhar]

Actually, infinite means unbounded

Sorry, Nikkoo, I missed your quote when I was looking for it. You will probably want to take issue with my previous post. :)

 

[my_wan]

As ObsessiveMathsFreak pointed out, this "depends on what you mean by infinite." In terms of mathematical infinities your assertion is undoubtedly true, but it is a mathematical "truth" and has no significance in reality.

I have already gone over how it is relevant to the physical sciences, by way of Hilbert space. You can call Hilbert space a mathematical fiction, or slide rule of sorts, used to calculate. But a rejection of mathematically defined infinities has very real empirical consequences. The inability to scale quantum computers being a major one. Others involves issues surrounding Bell's theorem, and other no-go theorems.

You cannot escape the issues mathematicians have worked around with a "mathematical fiction" clause. Finite mathematics is inconsistent without mathematical infinities. How do you propose the reinstate consistency if you reimpose a finiteness condition on the physical world? You can't simply close your eyes and pretend it has no physical consequences, whether those consequences ultimately justify actual infinities or not.

 

[Nessdude14]

Infinity is a number that cannot be divided, cannot be measured, and cannot be contained. This infinite universe obviously does not exist due to the fact that all pieces of space are made of this ultimately small unit.
Every quantity can be described by this unit, thus making the concept of infinity null.

That's an absurd assertion. The number line is infinite, but we can still use integers to measure and divide segments of the line.

 

[bahamagreen]

This can't be justified. The set of all actual hotel customers anywhere in the world does not comprise all possible hotel customers. If that was necessarily true then in order for hotels to have any customers they must have every person on Earth as a customer. Conversely, by this logic, since I am not a customer, either hotels have no customers or I am not a potential customer. Neither of which is true.

How about addressing my previous post #28 first?

 

[my_wan]

How about addressing my previous post #28 first?

I'm suggesting that is a flaw because you are defining some persons as both a non-customer and a customer - because the naturals and reals share some members in common (all naturals are members of the reals, some reals are members of the naturals).

Yes, that is why I specified multiplying by an irrational, which is a subset of the real numbers but not a subset of the natural numbers. Hence the symmetry is complete.

I can also relabel all natural numbers as real numbers simply by multiplying their name tags with an irrational number and assigning them that number.

Recap:
Set of all people (customers and potential customer) = real numbers
Set of all actual customers = natural numbers
Subset of non-customers (potential customers) = irrational numbers

Given that the set of all potential customers is larger than the set of all actual customers, whether irrationals are contained in the set of reals or not, there remains more potential customers than actual customer. I specified an irrational for the explicit purpose of of avoid a clash with your naturals (set of actual customers), in spite of the fact that there are more reals than naturals, which technically mooted the rebuttal anyway.

 

[my_wan]

bahamagreen,
I can sympathize with your difficulty on the hotel paradox. I have tried thinking through possible ways of getting around it. All of which involve refining definitions more than the paradox makes explicit. I'll try to construct a version of your argument that is harder to deconstruct. Though I will not offer any proof either. Neither does it reject infinities.

For instance, if you compare the statements:
(1) Hilbert's hotel contains an infinite number of rooms in which each room contains an occupant.
(2) Hilbert's hotel contains an infinite number of rooms and occupied by an infinite number of guest, for which the cardinal numbers are equal.

The hotel paradox essentially assumes these statements are equivalent. I suspect that this is not fully justified. There is only one countably infinite cardinal, \aleph_0, but there are uncountably many countably infinite ordinals ω. By definition in statement (1) we have assigned a one to one correspondence between the number of occupants and the number of rooms. Thus the one to one correspondence is in reference to ordinals rather than cardinals.

Now the equivalence of the above 2 statements is predicated on the fact that ω + 1 = ω, i.e., addition and multiplication are not commutative. Seems straightforward enough, just as 0*1=0 and 1 + 0 = 1. However, if we look to calculus, 0 may not equal 0, but rather an infinitesimal ΔL, the inverse of an infinity. In calculus we must make use of these limits specifically to avoid these self same ordinal properties we associate with 0, and inversely infinity. If calculus requires us to avoid this property with respect to zero, why is 1/ΔL special? ΔL simply has the equivalence class of 0, wrt a finite interval.

This wouldn't change much mathematically in operational terms, but would allow us to make a distinction between statements (1) and (2). In both ΔL and 1/ΔL the only thing that changes is the ordinal, not the cardinal. This would dictate that if the ordinal by definition has a one to one correspondence then there simply is no room to add another guest, though the cardinal remains the same up to \aleph_1.

We can also still accommodate more guest, when ω_1 = ω_2, under the condition that switching rooms requires some finite time interval, or a time interval with a cardinal number less than the cardinal number of guest.

Can anybody destruct that argument? It would be interesting to try and prove also.

 

[Endervhar]

How do you propose the reinstate consistency if you reimpose a finiteness condition on the physical world?

I am not trying to impose a finiteness condition on the physical world. In fact I would reason that the cosmos (= everything that exists) must be infinite, otherwise we would not be here. The Universe may be finite; I believe there are very good arguments that it is; but this is simply the result of our limited perception. Here I am not talking about the limits of our equipment, but the fact that we observe everything within the restrictions of our 3+1 dimensions, which must not match the dimensionality of the infinite cosmos. Think of a spider walking through Flatland.

 

[KUMAR SIDHANT]

i agreed that universe is expanding regularly.. that's why we can't get the actual shape of it dude

 

[InfiniteNiran]

Something happens before something and something happens after something. It means there is always beginning for the beginning and there is always beginning after ending according to Thermodynamics law. Universe, Universes, Multiverses, Infiniverses and etc are incomprehensibly Infinite ∞

 

[bahamagreen]

It's been a little while, but I think I know what my problem is with Hilbert's Hotel.

The problem posed in the story is that the infinity of rooms are each already occupied. So the question is how to assign a room to the new guest...

I'm seeing an equivalence between the new guest and his new room. In looking at how both the guest and the room might be potential members of their infinite sets, the additional guest is allowed to exist and show up, but the additional room is not allowed to exist and thereby causes the assignment problem for the hotel manager...

Why is it that in spite of an infinite number of guests already assigned to rooms, another guest is allowed to exist, yet of the infinite number of rooms, another one is not allowed to exist and be found?

Another way to look a this is to break the thing into two independent questions:

1] Given an infinite hotel of rooms, are there any additional ones out there? Hilbert says, "No"...
2] Given an infinite world of guests, are there any additional ones out there? Hilbert says, "Yes".

From that difference he presents the paradox of solving the match up of guests to rooms... but why the two answers to the same kind of question? That is the premise flaw I see here...
Unless I'm still missing something.

 

[mfb]

Why is it that in spite of an infinite number of guests already assigned to rooms, another guest is allowed to exist, yet of the infinite number of rooms, another one is not allowed to exist and be found?

You can look at a different setup where an additional guest shows up and an additional room is built, but that is trivial to solve.

Hilbert does not say "there is". The question is "imagine that, ... , how can we solve it?".By the way: To make room for the additional guest, an infinite number of guests have to move. This is certainly annoying for them. And "a bit annoying" for an infinite amount of guests is worse than "very annoying" (sleeping in the corridor) for one guest ;).

 

[bahamagreen]

"The question is "imagine that, ... , how can we solve it?"."

If that is the case, then it is solved by noticing that the premise is based on a clear logical inconsistency.


If you imagine a hotel with infinite rooms, you have to apply the same logic to imagining an infinite population of guests... they are equivalent and need to be treated identically when considering the existence of an additional element.

The point of the premise is that two different conditions are being applied to two logically identical objects, one condition allows no new elements and the other does allow new elements.

Basically, the premise includes a guest without a room showing up, then more guests, then bus loads of guests, etc. If there are infinite guests already in the rooms of the hotel, and more guests are allowed to appear, then the same applies to the rooms; there are an infinite number of rooms, but more can be found.
To say no more rooms can be found is the same as saying no additional guests can appear.

"Infinite" may or may not entail "all", but either way needs to be applied to both the rooms and the guests...
If infinite means "all", then the infinity of guests in rooms already means no additional guest can show up. If infinite does not mean "all", then there is at least one additional unoccupied room in the hotel.

 

[mfb]

@bahamagreen: That does not make sense.

Imagine I have 3 bananas and give them to 3 monkeys. Each monkey gets a banana.
Imagine I have 4 bananas and give them to 3 monkeys. You can imagine that, right? Even after I added a banana.

It is an imaginary situation, I can use any numbers I like.
I can distribute infinite bananas on 3 monkeys - at least one monkey has to get an infinite amount of bananas, so what?
I can distribute infinite bananas on an infinitely many monkeys. And then I can take another banana and give it to monkeys.

 

[Dead Boss]

Basically, the premise includes a guest without a room showing up, then more guests, then bus loads of guests, etc. If there are infinite guests already in the rooms of the hotel, and more guests are allowed to appear, then the same applies to the rooms; there are an infinite number of rooms, but more can be found.
To say no more rooms can be found is the same as saying no additional guests can appear.

It doesn't matter whether new rooms can be found or not, because you can comfortably fit the new guests in the rooms that are already occupied.

 

[bahamagreen]

You're not getting the point.
Forget about the methods of putting extra guests into the rooms.

Look at the facts of the problem:

Hotel has
Infinity of guests
Infinity of rooms

If you allow that an additional guest (without a room) can exist,
you must also allow that an additional room (without a guest) can exist.
These two things are the logically identical, therefore the premise that the new guest has no room is false.
If a hotel with infinity of rooms does not have an empty room available, then likewise, with an infinity of guests with rooms, there will not be possible a new guest without a room.

Look at it this way; what if the original paradox had been reversed?
The hotel has an infinity of rooms and guests...
Then one day a new empty room is discovered, but the manager needs to report full occupancy to be paid his bonus.
So he shuffles the guests through the rooms (including the new one).
The simple solution is for anther guest to appear (like in the original).

See, the discovery of a new guest is just like the discovery of a new room.

 

[dyb]

If the universe were infinite and existed for an infinite amount of time and the universe weren't expanding, and the universe had the same kind of distribution of galaxies everywhere, then night would be bright as day because no matter what direction you look there would be light coming from that direction.

But, due to red-shift, far away galaxies may no longer appear in the visible spectrum (but you could measure it). And if the Universe expanded fast enough outer parts could become "disconnected". So it's a little harder to figure out.

 

[Endervhar]

Bahamageen, it’s refreshing to find someone who thinks along the same lines as I do about infinity.

I believe the main problem is that since Cantor “tamed” infinity there have been two distinct perspectives; the mathematical infinity and the real infinity. Much confusion arises because there is such a profound difference between the two, yet we can discuss infinity with one person using one form and the other person using the other form.

Mathematical infinities are simply mathematical concepts that are of value in some calculations. There exists an infinite number of mathematical infinities which can be of different sizes.

Your arguments apply to real infinity. There can be only one, and it must include everything. If you try to introduce numbers to such an infinity your calculations lead to nonsensical answers.

. Take for example the (UK) national Lottery. In an infinite universe, an infinite lottery becomes possible, and therefore inevitable, not only that, it must occur an infinite number of times. So, what would this infinite lottery be like? There would be an infinite number of people taking part, the staked money would be infinite, therefore, the jackpot (being a percentage of the stake) would also be infinite, the jackpot winners (being a percentage of the infinite number of people taking part) would be infinite, as would the number of losers. We can see from this that an infinite number of people would win an infinite share of an infinite amount of money, but, paradoxically, the same infinite number of people would not be winners at all.

Mathematicians can find their way, logically, through Hilbert’s Hotel, followed by infinite guests grumpily changing rooms; but in the real world it calls to mind the well known debate about angels and pins.

 

[Dead Boss]

You're not getting the point.
Forget about the methods of putting extra guests into the rooms.

Look at the facts of the problem:

Hotel has
Infinity of guests
Infinity of rooms

If you allow that an additional guest (without a room) can exist,
you must also allow that an additional room (without a guest) can exist.
These two things are the logically identical, therefore the premise that the new guest has no room is false.
If a hotel with infinity of rooms does not have an empty room available, then likewise, with an infinity of guests with rooms, there will not be possible a new guest without a room.
.

You're not getting the point that it's not important whether there is a room for the guest or not: you don't need it even if there is. You can fit the new guest into one of the old rooms and not use any new room. The premise that there is no room for the guest is not a premise of the mathematics of the paradox but only of the "story" behind it. The simple mathematical idea is wrapped in a textual anecdote and that anecdote requires the "no room" premise for the problem of fitting the new guest to exist. You can just as well state that the hotel manager is a freak who loves to move guests around. Nothing changes the fact that you can find a bijective function from integers to naturals (for example) which is all this "problem" is about.

 

[bahamagreen]

I understand it's just a story... but the "story" has a flaw, an inconsistent logical treatment of the rooms vs the guests.

It makes no sense to allow for a "new" guest to be found and them maintain that a "new" room can't be found.

I understand that IF you take the first statement as a given, THEN yes, the story acts fine as a puzzle for how to place the new guest in a room, in spite of all the infinite rooms already being occupied by the infinite guests.

The problem with this story is that anyone who thinks about the origin of the new guest must logically conclude that the first statement is inconsistent... that statement being that no new rooms can exist.

I'm pointing out a separation between two things:

The story as presented (which may be fine for setting up how to solve a problem with infinities)
and
Looking at the problem on its face and realizing that it is a false problem.

The reason I think this is important is because if the question (story) is allowed to contain illogical inconsistencies, what rules then prevent the answer from also using logical inconsistencies?
If the set up for the story is "wrong", then how is any answer "right"?
Or how can any answer that just magically and illogically "solves" the problem be legitimately denied?

I expect the rules that apply to evaluating the answer to be the same rules that must apply to the construction of the question.
Hilbert's Hotel does not meet that expectation; the problem needs to be logically repaired before being asked, and after the repair it is no longer a problem, no paradox.

For learning purposes, maybe the problem should be altered so as not to raise the question about where the new guest came from by having the hotel originally occupied by an infinity of women, and the new guest is a man (a lucky man!), then do the subsequent movements of occupants to insure one person per room is enforced... (not so lucky man).

 

[dyb]

Perhaps an analogy you will understand:

Consider the number 10/9. It is 1.1111111... with an infinite number of 1's after the decimal point. Now you may argue that 1/9 is the 10th of 10/9, so the decimal digits get shifted by one to the right and therefore there must be one additional 1 after the decimal point. So, you could argue that 10/9-1/9 would need to be slightly below 1. But it's not. It's exactly 1 because 10/9-1/9 = 9/9 = 1.

(This also follows from the definition of decimal using limits)