disproving infinity paradoxes; Hilbert’s Hotel?

[Amorphos]

disproving infinity paradoxes; Hilbert’s Hotel?

here’s the old infinite-hotel puzzle (also known as “Hilbert’s Hotel”) In a hotel containing infinitely many rooms, all of which are full, how do you find room for infinitely many new guests? Simply move every guest to the room with twice the number - room 1 moves to room 2, room 2 to room 4, 3 to 6, and so on - and then all the odd-numbered rooms are free.

every room is ‘full’? what we are dealing with here is a problem of spaces and occupations, we can move a guest out of a given room to the next to create room. however if each room is full - literally - let us say that you have an infinite amount of rooms and each is filled with a wooden block the size of the room, then you cannot move any of the occupants to anew room! the puzzle then is simply set on unsound foundations, we are dealing with finite amount being moved around an infinite amount of locations.

secondly, we cannot have an infinite amount of rooms, this is the same as having an infinite amount of numbers - we cannot, we may keep counting all we like yet never get any closer to an infinite amount.

this and other arguments may be used to show how we cannot have an infinitely cyclic universe, and how we cannot have this universe or any given limited energy entity as infinite in any way! this implies that we cannot have an infinite amount of universes too.

[Simetrical]

however if each room is full - literally - let us say that you have an infinite amount of rooms and each is filled with a wooden block the size of the room, then you cannot move any of the occupants to anew room!

Why not?

secondly, we cannot have an infinite amount of rooms, this is the same as having an infinite amount of numbers - we cannot, we may keep counting all we like yet never get any closer to an infinite amount.

We do have an infinite number of numbers. We cannot write them all out, and likewise obviously the Hilbert Hotel cannot exist in reality. Infinity is a completely valid and critically important mathematical concept, necessary among other things for calculus (which is indispensable to physics and engineering, or at least has historically been so). The Hilbert Hotel is an analogy designed to illustrate that infinity can act counterintuitively. That it cannot exist physically does not diminish its correctness in mathematics, which deals with things that do not exist physically (and then not infrequently applies the results to things that do exist physically).

this and other arguments may be used to show how we cannot have an infinitely cyclic universe and how we cannot have this universe or any given limited energy entity as infinite in any way! this implies that we cannot have an infinite amount of universes too.

That's a non sequitur. It's quite possible that the universe is infinite in temporal extent. That you cannot imagine such a thing does not negate the possibility. You can never prove or disprove empiricals logically: they must be proven empirically (and likewise you can't argue for or against logical constructs such as Hilbert's Hotel via empirical arguments). Logic alone cannot possibly demonstrate even the slightest thing about the universe.

As for finite energy, that is again an empirical, not logical, question. It has been assumed that there's a finite amount of energy in our universe, based on available evidence, but first of all that may be ultimately demonstrated false; and second of all, it does not mean that there can't be infinite energy spread across infinite time or infinite universes. We only have reason to believe that there is finite energy in our universe, and only at any given time.

[chris_uk_83]

Your entire argument rests on word games and a lack of understanding of the concept of infinity, which is a concept, not a real thing as Sim so eloquently put it.

[Curtana]

There appears to be an infinite supply of bull****. Therefore the infinite exists.

[The Sundance Kid]

hahaha, another cogent scientific debate falls prey to the twisted logic of someone who couldn't care less. Congratulations, Curtana

It took mankind long enough to work out that 0 was a value, so maybe it'll take as long again for us to come round to the concept of infinity.

[Curtana]

Sorry to go off topic again but resistance is futile.
Actually it was the Hindus who invented 0 precisely because their religion talked about infinite possibilities. Early texts describe how Shiva can lay hair upon hair upon hair and so on ad infinitum..... The number system we know and love and its 0, 10, 20, was a direct result of Hinduism.

[Chaigidel]

doesnt 0 represent infinite in both a logical and mathematical sense?

[Amorphos]

sim

Why not?

every room is full! its volume is completely taken up and so too the volume of the next room.

We do have an infinite number of numbers. We cannot write them all out

we potentially have an infinite amount in real terms, an integer is simply a metaphor that represents a given item or example. sure you can have an infinite amount of metaphors, but they are not real, when we apply them to reality we only ever have a finite or limited amount. i agree that potential infinities are crucial to certain areas of science, my reasoning is that we should understand the difference between potential infinities and an actual infinity.

That’s a non sequitur

?
eh!
Main Entry: se·qui·tur
Pronunciation: ‘se-kw&-t&r, -”tur
Function: noun
Etymology: Latin, it follows, 3d person singular present indicative of sequi to follow -- more at SUE
: the conclusion of an inference : CONSEQUENCE

what do you mean exactly! i can only presume you are saying that the limits of our dimension discludes the understanding of the infinite.

You can never prove or disprove empiricals logically

why on earth not. one may solve mathematical problems with mathematics, infinity is no different - we cannot test it can we!


chris uk

Your entire argument rests on word games and a lack of understanding of the concept of infinity, which is a concept, not a real thing as Sim so eloquently put it.

i think these are reasonable examples, i have read many books on it and discussed it at length with my brother whom is a very advanced physacist, and various profs including the one at a philosophy workgroup i go to. i think in truth it is others whom don’t understand it!

http://www-groups.dcs.st-and.ac.uk/~.../Infinity.html

http://en.wikipedia.org/wiki/Infinity

it is a reasoned argument that you cannot disprove so you resort to evasive tactics. sim just argued that energy may be infinite - so why are you so sure it is not. you can’t have it both ways, using one argument where it suits you then another where it does not. for me infinity has to exist - anything else is illogical.

why not simply understand the problem correctly.

melboy

absolutely

curtana

the hindu idea of infinity:

The Isha Upanishad of the Yajurveda (c. 4th to 3rd century BC) states that “if you remove a part from infinity or add a part to infinity, still what remains is infinity”.
Pur?am ada? pur?am idam (That is full, this is full)
pur?at pur?am udacyate (From the full, the full is subtracted)
pur?asya pur?am adaya (When the full is taken from the full)
pur?am evavasi?yate (The full still will remain.) - Isha Upanishad

interesting but questionable.

[Juvenal]

It would be wise to listen to Simetrical the Sage.

Hilbert's Hotel is an illustration of the counter-intuitive nature of infinity. Since there are no infinities in the world we perceive, it should not be suprising that the consequences of infinity defy common sense.

The ability to add an infinite number of guests to Hilbert's Hotel demonstrates that Infinity + Infinity = Infinity (which is of course counter-intuitive).

A good way to understand Infinity is to think of it as a process rather than a result. Thinking of Infinity as a process - i.e. that you can carry on finding occupied rooms indefinitely, it is easy to see that the paradox does in fact work.

every room is ‘full’? what we are dealing with here is a problem of spaces and occupations, we can move a guest out of a given room to the next to create room. however if each room is full - literally - let us say that you have an infinite amount of rooms and each is filled with a wooden block the size of the room, then you cannot move any of the occupants to anew room! the puzzle then is simply set on unsound foundations, we are dealing with finite amount being moved around an infinite amount of locations.

• All occupants are told simultaneously to move to the room with double the number (there are after all an infinite number of them, and we haven't got all day).
• Each guest will find the destination room vacant, because its occupant will already be moving to the next doubling of room number and so ad infinitum.
• Because there are an infinite number of rooms, every guest always has a destination room to move to, no matter how large their room number, and the guest they are replacing will also have a destination room and so will be able to vacate.

This works just as well with wooden blocks, except that you need an infinite number of porters to move them.

secondly, we cannot have an infinite amount of rooms, this is the same as having an infinite amount of numbers - we cannot, we may keep counting all we like yet never get any closer to an infinite amount.

Of course we cannot have an infinite amount of rooms - it is just an illustration, just like the Schrodinger's Cat paradox. This doesn't mean we might not have an infinity of something else - space for instance.

this and other arguments may be used to show how we cannot have an infinitely cyclic universe, and how we cannot have this universe or any given limited energy entity as infinite in any way! this implies that we cannot have an infinite amount of universes too.

The argument presented does not prove or disprove the reality of infinity. Since we don't know what started the Big Bang, I don't see how we can make any strong assertions about the nature of a cyclic universes.

Infinity may not be real at all. We cannot directly determine that something is infinite unless it is a mathematical concept, otherwise an infinite amount of counting would be required before you could come to any conclusion. For infinity to be real, we are therefore dependent on the belief that mathematics can describe reality.

PS. in case no one has explained it yet (I have been typing this post for a long time) - non-sequitur is usually used to describe a statement that has no connection with the current subject, but which has been used as if it did have. It means literally "does not follow".

[The Fish]

curtana

the hindu idea of infinity:

The Isha Upanishad of the Yajurveda (c. 4th to 3rd century BC) states that “if you remove a part from infinity or add a part to infinity, still what remains is infinity”.
Pur?am ada? pur?am idam (That is full, this is full)
pur?at pur?am udacyate (From the full, the full is subtracted)
pur?asya pur?am adaya (When the full is taken from the full)
pur?am evavasi?yate (The full still will remain.) - Isha Upanishad

interesting but questionable.

Quetz, this quote is an application of the mathematical concept of infinity to the quality of Brahman. As such (as people have pointed out previously) it is obvious that it does not represent a physical quantity, but an abstraction. Just as infinity isn't just a really big number, so too is Brahman not just a really large entity.

[Amorphos]

goaty

I think the problem is you are treating infinity like a large number (100 trillion, billion or something like that) which has a finite ending

the exact opposite actually.

the reason you say there wouldn’t be any rooms free because the last guy on the line doesn’t have anywhere to go

there is no first or last guy, its just that ‘all’ [irrespective of amount] rooms are ‘full’.

If there are an infinite number of occupied rooms and each occupant is to move to the room double its number then that moving will be going on forever

it is infinitely full though. here you would be presuming that it is part empty.

Its an endless sequence, there will always be a room being emptied as the one double its number is being filled

so 1 moves to 2, 2 moves to 4, yet both 2 and 4 are occupied, add infinitum. the paradox says that 2 would move out so that 1 could move in, yet where would 2 move to.
in short you have to have a space or indeed an infinite amount of spaces to move into, yet contrarily, to begin with the paradox states that all rooms are full.

paradox broken! as infinity paradoxes always can be.

fishy

Quetz, this quote is an application of the mathematical concept of infinity to the quality of Brahman.

i am saying brahman is incomparative [as is infinity as a real entity] - think on it my friend, it is one of the worlds great truths.

As such (as people have pointed out previously) it is obvious that it does not represent a physical quantity, but an abstraction

read my previous post or two, i hope it will show that this is certainly not what i am thinking

Just as infinity isn’t just a really big number, so too is Brahman not just a really large entity.

sure, infinity is ‘sizeless’ - without end nor edges - is not a very big thing. i don’t know why people presume i don’t know that already?

i found this most intriguing:

‘when the full is taken from the full, the full will still remain’

imagine an infinite ocean, if you extrapolated an infinite ocean from it what do you have left = 0. on another level, you cannot take anything from infinity as it cannot be broken! [i.e. into pieces!].

i believe that when the sages of ancient india had such realisations, it was concerning entities which had no value [actual] and were not physical. then you may have say; an aether of infinite yellow and another of red, just as you could have a spiritual nature e.g. ‘male mind’, and extrapolate anther infinity of the spiritual nature ‘female mind’. here we see an infinity taken from infinity, yet it is rather like or exactly like 0 taken from 0, as such entities are valueless in terms of physical reality - are metaphoric entities.

[DisgruntledGoat]

it is infinitely full though. here you would be presuming that it is part empty.

No I'm not. I'm presuming there are an infinite amount of rooms which are infinitely full. And if every double room (which goes on forever since there are infinite amounts of rooms) moves then there are at the same time an infinite amount of full rooms and an infinite amount of empty rooms. Which is why you can add infinity to infinity the precise point of the hotel analogy.

Let me try and explain it again. 1 moves to 2, 2 to 4, 4 to 8. Each room in front will be emptied because there will ALWAYS be a room to be emptied because there are an infinite number of rooms. Infinity will move to 2 times infinity and 2 times infinity will move to 4 times infinity.

the paradox says that 2 would move out so that 1 could move in, yet where would 2 move to.

Just as I said, it would move to four since four has moved to 8 and 8 to 16. All along the line and because there are an infinite amount of rooms occupants will keep on moving up double their room forever therefore ensuring that there will be an infinite amount of unoccupied rooms leaving room for the infinite amount of new occupants.

[The Sundance Kid]

melboy

absolutely

wait one second, did someone smarter than me just agree with me!? *faints*

[Amorphos]

juvinal

great post mate!

Infinity + Infinity = Infinity

if you understand infinity properly then you will find that your statement is like saying 0 + 0 = 0, when really nothing has occurred. Hilbert’s hotel is a false logic.

A good way to understand Infinity is to think of it as a process rather than a result. Thinking of Infinity as a process - i.e. that you can carry on finding occupied rooms indefinitely, it is easy to see that the paradox does in fact work.

a process it most certainly isn’t [ultimately anyhow], in fact it is the most non-process there is. processes are made up of operations where events occur ‘amounts’ are calculated, none of which happens with infinity - it just is. what you are describing here is infinite potential and continuance! you can keep finding occupied rooms yes, however as with numbers you never actually reach infinity you just keep climbing a hill that never ends.

Each guest will find the destination room vacant, because its occupant will already be moving to the next doubling of room number and so ad infinitum.

there would be no movement at all! each room is full so you cannot move from one to another as the next will always be occupied, it is a false logic.

Because there are an infinite number of rooms, every guest always has a destination room to move to, no matter how large their room number, and the guest they are replacing will also have a destination room and so will be able to vacate.

an infinite number of rooms with an infinite number of occupants. if each occupant fills the room there are no spaces left. if occupants are less than the volume of the room then we are simply switching variants of finite volumes between rooms, and sure you can do that all day it shows nothing.

Of course we cannot have an infinite amount of rooms - it is just an illustration, just like the Schrodinger’s Cat paradox. This doesn’t mean we might not have an infinity of something else - space for instance.

hmm i see that you could have say an infinite black space and an infinite yellow space, then red then orange etc [let us imagine that there are infinite colours or shades thereof]. if you had an infinite amount of occupants e.g. black occupants, yellow, red etc, then the paradox would work. what we are missing is that we cannot have an infinite quality whatsoever e.g. an infinite yellow, black or whatever. this is where we may divide true infinity from ‘partial’ infinities - and in that term alone shows the flaw.

...as we can have infinite qualities then at what point do we stop adding, everything is an ‘it’ of some description, all qualities all meanings all holistic entities etc. this is why people often arrive at god, they simply add everything into one place. if you want to explore all occupants of all possible infinities with infinite variation simply ‘define ‘it’.

there are many problems with infinite variability - another thread perhaps.

... It would be wise to listen to Simetrical the Sage

you taught me well obi wan, now i am the master.

[ajm317]

if you understand infinity properly then you will find that your statement is like saying 0 + 0 = 0, when really nothing has occurred. Hilbert’s hotel is a false logic.

There is nothing wrong with the logic. It just describes an impossible situation (a hotel with an infinite number of rooms.) That you get nonsensical results is not particularly surprising.

a process it most certainly isn’t [ultimately anyhow], in fact it is the most non-process there is. processes are made up of operations where events occur ‘amounts’ are calculated, none of which happens with infinity - it just is. what you are describing here is infinite potential and continuance! you can keep finding occupied rooms yes, however as with numbers you never actually reach infinity you just keep climbing a hill that never ends.

In fact a process is a good way of describing infinity. To say something is infinity ultimately means little. There are many kinds of infinities. For example, there is an infinite number of rational numbers, and there is also an infinite number of irrational numbers and yet it can be proven there are more irrational numbers than there are rational ones. All infinities are not equal. It is important to know HOW a value gets to infinity. Does it approach it linearly? Or exponentially?

there would be no movement at all! each room is full so you cannot move from one to another as the next will always be occupied, it is a false logic

Again, nothing wrong with the logic. You just can't have an infinite hotel.

That however does not mean you cannot have an infinite space, or infinite time. That such a thing would have unusual consequences does not particularly bother me. Physics is full of unusual consequences. Open a Quantum textbook sometime.

As has been said nothing empircal can be proven or disproven with logic alone. This was the mistake of the Greeks. The only way to determine if the universe is infinite or not is to measure it. Of course, you can never get infinity as a measurement, but if your measurements approach infinity to within experimental error you may assume it to be so, afterall you cannot tell the difference anyway. That an infinite universe may lead to unusual consequences is not the universes concern.

[Amorphos]

ajm317

some good arguments there!

For example, there is an infinite number of rational numbers, and there is also an infinite number of irrational numbers and yet it can be proven there are more irrational numbers than there are rational ones.

no, there is an infinite number of metaphoric examples! now name an infinite amount of non metaphoric examples e.g. if you added up every single particle that exists or has ever existed, would it be infinite - no, because energy is conserved there can only ever be a finite amount of them. btw i am aware of virtual particles, this doesn’t detract from the example as they too are not infinite in amount.

It is important to know HOW a value gets to infinity. Does it approach it linearly? Or exponentially?

yes how do they get there indeed? however it is approached you have to arrive at an infinite number. let us imagine a non linear example; 1 to infinity, note the end point i.e. the beginning. imagine that we have an infinite set, we would also have an infinite remaining, for example we could count from 1 inversely to infinity [-1 -2 etc]. regardless of approach we always have an infinity remaining.
now let us say that you have an infinite amount of ‘it’s or ‘x’s, so as to avoid numerical or other values; we may visualise this as say points, there would always also be an infinite amount of comparative space.

infinity as actual is incomparative! this is the most important aspect of boundlessness - we cannot have a bounded amount as it will never compare. this includes any ‘amount’ of anything.

Again, nothing wrong with the logic. You just can’t have an infinite hotel

its an analogy. we are actually describing entities; that of volumes and occupants, ultimately points and spaces. here then i would be saying that if you have an infinite amount of both then none may be added.

As has been said nothing empirical can be proven or disproven with logic alone
empirical:
Definition: 1. based on observation and experiment: based on or characterized by observation and experiment instead of theory

yet we cannot actually make an absolute observation!? so what you are saying is that we can only say; ‘this is so’, when we make an inexact observation of it, and that if logically i said 2 + 2 = 4 i would actually have to have 2 things and 2 more to show this is true. yet we can say that there is no such thing as 2 things as being in the class of two things, that is to say; two holistic objects are two distinct entities and cannot rightly be counted as 2 any more than any number of holistic objects may be. the same goes for particles and ultimately for their improbability, that there are no distinct entities in existence thus to be regarded as 1 + 1 entities.

in short we may only show that 2 + 2 = 4 by logic alone!

That an infinite universe may lead to unusual consequences is not the universes concern

it is if we are saying that the universe is infinitely cyclic. which is the current argument against there being a beginning.

[ajm317]

no, there is an infinite number of metaphoric examples! now name an infinite amount of non metaphoric examples e.g. if you added up every single particle that exists or has ever existed, would it be infinite - no, because energy is conserved there can only ever be a finite amount of them.

1. An infinite number of particles does not imply infinite energy. The total energy content of the universe is, according to the best observations, 0. The universe also contains negative energy in the form of the gravitational potential energy.
2. You are assuming there is a finite amount of energy in the universe. This is not necessarily true.

yet we cannot actually make an absolute observation!? so what you are saying is that we can only say; ‘this is so’, when we make an inexact observation of it, and that if logically i said 2 + 2 = 4 i would actually have to have 2 things and 2 more to show this is true.

No, 2+2=4 is math. Math may always be proved by logic alone as it is a purely rational contruct created by man.

two holistic objects are two distinct entities and cannot rightly be counted as 2

Actually quantum particles are indistinguishable (if I understand what you're saying, which is no gaurantee).

it is if we are saying that the universe is infinitely cyclic. which is the current argument against there being a beginning.

Yes, but fortunately an unusual demonstration of the concept of infinity on a hotel does nothing to disprove a cyclic universe.

The Greeks played such logical games all the time. They accomplished nothing, because they did no experimentation. Take for example Plato's concept of the prime mover. The underlying assumption was that all actions have causes, something which quantum mechanics has shown to be false. Or take Zeno's paradox. He used logic to "prove" that nothing ever moves! Newton proved him wrong using calculus.

Our ability to understand the universe using logic is limited. Do not rely on it too heavily. The conerstone of modern science, that which seperates us from the Greek philosophers, is experiment, not theory.

[DisgruntledGoat]

Okay I've read, and re-read your first post trying to figure out what about infinity you are misunderstanding. I think the problem is you are treating infinity like a large number (100 trillion, billion or something like that) which has a finite ending, the reason you say there wouldn't be any rooms free because the last guy on the line doesn't have anywhere to go. But as others have mentioned before infinity isn't a number or anything tangible at all. Its a concept and a fairly simple one (I believe). Simply put, infinity is to go on forever with no end. If there are an infinite number of occupied rooms and each occupant is to move to the room double its number then that moving will be going on forever. Because the room change is going to occur forever and never stop it cannot be attached to anything physical by saying "they would simply run out of room." It can't happen as there isn't any room to run out of. Its an endless sequence, there will always be a room being emptied as the one double its number is being filled. Hopefully I haven't confused anyone. I guess since some of use have been immersed in these kinds of concepts for so long that it is tough for us to fully express our understanding of it.

[lt1956]

Interesting all the rooms are full and yet there are more people coming to fill them yet infinity has no beginning and so the new people cannot exist they would simple be "there".

Interesting, I see BOTH sides to this, funny thing is can both sides be Right and Wrong at the same time, as this infinity question is? lol

Love the discussion but let me know if you ever get the asnwer. ;-) Meanwhile give SPQR mod for RTW a try, I think there are infinite amount of armies the AI makes..... lol

[Juvenal]

Really LT - that was a truely shameless plug.

It is lucky that I have an infinite amount of respect for your modding abilities or I might have considered reporting your post (and probably getting myself expelled for my troubles).

PS. I hope that you are not in the habit of telling a combatant locked in a (figurative) struggle to the death with his nemesis that you...

see BOTH sides to this