disproving infinity paradoxes; Hilbert’s Hotel?

[_Z_]

disproving infinity paradoxes; Hilbert’s Hotel?

i thought i would have a crack at this ole chesnut, see what you think?
here’s the 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.
however, 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 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.
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god and infinity, or nirvana perhaps?

i am beginning to think that there is a parallel between the notion of where i am heading here [1] and a rather more ancient one [2]:
1. infinity is incomparative - for me this is one of life’s great truths.
2. "no that he is never in anything" a reference to god and his separateness from the created world.
many people especially in medieval times, have associated god and infinity, do you,and how do you connect them? and would you think this is appropriate? or would you separate god from infinity.

 

[wil]

If it is full of your block of wood (rather than guests) it could be equally full as it stands, full of space filled with gas, oxygen, nitrogen etc...and plenty of room for an infinite number of guests.

 

[_Z_]

good point wil. what i was saying is that we are imagining guests moving freely about, that they may move from one room [imagined with no corridors] to the next. however the point is that all rooms are full, hence by using extra sets we may not find more rooms; infinity = all.

the point is that you cannot have a finite amount of infinite. you cannot have mathematics of infinity, you can only have mathematics of potential infinities. the cardinality of numbers and sets etc make absolutely no difference to this. what we are actually describing is a potentially full hotel i.e. all the natural numbers from 1 to X are used. it is then true to say that you can have other kinds of sets of numbers running parallel or tangental to this.
what we need to do is visualise a real infinity, then we see where the maths of potential infinities doesn’t touch it. a real infinity would necessarily be whole and complete, it would be everything at zero value.
just as we can see polarised energy become zero at its neutral point, we may perhaps imagine every aspect of reality as belonging to this universal balance or as having a zero value at their base which would be infinity.

a quote from a friend of mine:
"If the universe was infinitely old, everything imaginable would have existed before and there could be no "here and now".

question:
If the universe is cyclic then it could get rid of everything that was and repeat?
1. if it does the same thing twice it would actually only be doing it once for example; if it were only composed of 0 and 1, then 1 could not repeat as it would have already happened in universal time, as if it is re-mapping itself.
2. also consider that perhaps.. the potential and cause would have occurred and become real, for the same thing to happen twice you need a second cause and another potential, in which case the entire event would not be exactly the same?
3. you also cannot have 1 X 1 recurring up to infinity in ‘real’ terms, just as you cannot have 12345679-infinity.

given that the universe cannot be infinitely repeating, then you are left with infinite variability...
in short if you can have one infinite set, then you must have all sets, i.e. if you have all numbers you have all everythings. if we had all universes, 1 for each number on that apparent infinite line then they would be squashed into an infinitesimal point, then we would have the dilemma of what is between the points - apart from the fact that our universe couldn’t exist as it is!
now think on the consequences of infinite variability, you would have to have every kind of universe imaginable no matter how crazy it is, it would have to occur! e.g...
one universe is made of candy floss with vampiric chocolate bunnies that hop from cloud to cloud biting innocent virgins and drinking their raspberry juice blood . another universe would be one that eats other universes, which in turn is eaten and so on and so forth, until we arrive at the ultimate predator universe which devours all universes infinitely quicky... poof they’re all gone before a moment has passed.

this is why i don’t believe in infinite variability! there is a similar dilema for infinite amounts of different planets, that is without even accepting that an infinite amount of planets would be infinitely compressed and hence not planets.

 

[wil]

then none of this could be infinitely thought provoking?

The one thing I find interesting about what ifs, is that they don't allow what ifs in the answer.

We have some possible potential paradox based on a hypothetical yet the hypothetical isn't allowed in the solution.

Can G-d make a rock so big he can't lift it?

 

[Neemai]

Can any equation, which includes infinity as a factor, ever be proven in a non-theoretical way?

... Neemai

 

[_Z_]

wil

then none of this could be infinitely thought provoking?

if it takes you there then you have just met god

The one thing I find interesting about what ifs, is that they don’t allow what ifs in the answer.

good point, although the notion; ‘what is infinity’ - which is the real question being alluded to here - has quite a bit of free ground to cover.

We have some possible potential paradox based on a hypothetical yet the hypothetical isn’t allowed in the solution.

‘infinity as incomparative’ or as itself beyond all approaches, is the hypothetical solution - isn’t it?

Can G-d make a rock so big he can’t lift it?

yet another good point! i’ll here inject some hindu philosophy on the matter...
quote: 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". i.e.
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

i think the understanding is that; if you remove an infinity from infinity, then infinity must still remain infinity - which is full. thus in these terms one may ‘take from the full, the full will still remain’
although it is full of emptiness? i.e. has a value of zero, where all things of the everything are balanced in the neutral or otherwise has a 0 value. thus taking the full out of the full is more taking space out of space.

Neemai, hi

Can any equation, which includes infinity as a factor, ever be proven in a non-theoretical way?

probably not, yet it may be alluded to or perhaps understood simply as an incomparative ‘entity’ [although it has no entity of entity]. is this not like any wisdom or gnosis, we may only understand it when we are there.

 

[China Cat Sunflower]

It would seem that there should be some differentiation in terms between intrinsic infinity: infinity of itself, cyclical infinity, and apparent infinity. A self-perpetuating closed system has an aspect of infinity, as does the output of a randomly generated effect.

Chris

 

[seattlegal]

yet another good point! i’ll here inject some hindu philosophy on the matter...
quote: 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". i.e.
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

i think the understanding is that; if you remove an infinity from infinity, then infinity must still remain infinity - which is full. thus in these terms one may ‘take from the full, the full will still remain’
although it is full of emptiness? i.e. has a value of zero, where all things of the everything are balanced in the neutral or otherwise has a 0 value. thus taking the full out of the full is more taking space out of space.

This is saying that infinity can be divided--the set of odd numbers is an infinite set, but not an all inclusive set. You can define an infinite set in such a "narrow" way that members of this subset might be comparatively rare, but still infinite. Applying this to the Hilbert's Hotel analogy, it could be that the guests you are trying to fit into the hotel just do not fit into the parameters of Hilbert's Hotel--{Hilbert's hotel infinite set might be odd numbers, while the guests you are trying to fit in might be even numbers. In this case, there would be no place for these guests to be found in the hotel, and all of the rooms would appear to be "full," due to the lack of "even numbered" spaces.}

however the point is that all rooms are full, hence by using extra sets we may not find more rooms; infinity = all.

Using the example listed above, one can see that infinite does not necessarily mean 'all.' That is where the fallacy is.

 

[Tao_Equus]

Seattlegal,

Using the example listed above, one can see that infinite does not necessarily mean 'all.' That is where the fallacy is.

I think thats the nub of it. Infinity is always a step beyond all.

I think the real challenge is not to attempt to comprehend the infinite but to set the limitation of finite space. That is a concept beyond me. Regardless of all warpings and loopings physicists dream up.

TE

 

[seattlegal]

Seattlegal,

I think thats the nub of it. Infinity is always a step beyond all.

I think the real challenge is not to attempt to comprehend the infinite but to set the limitation of finite space. That is a concept beyond me. Regardless of all warpings and loopings physicists dream up.

TE

Wouldn't that than make the term 'infinity' into a 'wildcard,' meaning 'any and/or all' within the given parameters? The parameters would then become the 'limits' on 'infinity.' (I suppose that the idea of setting limits on infinity seems like an oxymoron to some. Without parameters, 'infinity' then is beyond description.) {Would that be anything like "the Tao that can be named is not eternal Tao?"}

 

[wil]

I love the various sizes of infinities...that some infinities are bigger than others...

 

[Tao_Equus]

Wouldn' {Would that be anything like "the Tao that can be named is not eternal Tao?"}

I think thats pretty accurate. You cant describe the indescribable. You cant encompass it intellectually because encompassing makes it finite. Neither can you reduce it, magnify it, dissect it, square it or anything else. It really is a futile exercise and one of these rare things genuinely beyond philosophy. Again I say the real challenge is in creating a finite universe.. and is nothing something?

TE

 

[_Z_]

china cat sunflower.

some definitions i use:

potential infinity; the process of counting to infinity, yet never actually getting there because you cannot have a finite amount of infinity, and ‘infinity may not be built up to’.
comparative infinities; infinity as compared to the finite [or the quantum].
incomparative infinity; infinity as actual. it remains continually in a state that cannot be compared to by any other means than itself. in a sense this means that infinity is only ever infinity, there are no amounts of it and there are not multiples thereof - as it is all, complete and boundless.
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infinity cannot be built up to:
i would put forwards the idea that you cannot have an infinite amount of natural numbers, you simply keep counting never getting there. each integer is finite in meaning, no matter how many you add they remain a collection of finite parts, to which cannot add up too or be built up to an infinite amount.
equally so; you have a set of infinite natural numbers, then you may add another set imagined as say parallel to the first. we could go on like that for ever yes? we could have any kind of sets we choose right, and keep adding! however if the first set of natural numbers were truly infinite then they would necessarily include all, so as to be a function of the infinite - or a truly infinite set. thence if natural numbers did add up to an infinite number set where do you find room for the next set i.e. all the places have been used.
thus we cannot actually have infinite sets, only potential infinite sets or imagined ones.

seattlegal, hi

about the concept of infinite odd and an infinite even number sets; what is missing? ...each from each! in this scenario we are imagining 2468, and 13579 to infinity [see above for the paradox this induces], but 1 and 3 are missing from 2 and 4 etc thus each set to be completed and made infinite would have to include each other so as to include 'all' where infinity is boundless, whole and all {?}.
the fallacy is where we are expressing infinity with finite entities [here being integers of natural numbers etc]. surely infinity may only be expressed as itself. a finite amount of infinite is paradoxical indeed even contradictory!


imagine it like a pool of water, you put a glass of water in the centre and the pool gets bigger [by displacement],you keep adding glasses and the pool keeps getting comparatively bigger.
no matter how many finite amounts you add together, you never reach infinity, in fact you can add 1,000,000,000,000,000,000,000,000,000,000 amounts and infinity remains equally greater as when compared to just 1.

tao equus

I think thats the nub of it. Infinity is always a step beyond all.

i don’t see how you can have more than all. no matter by what guise the idea is arrived at.
secondly infinity is only ‘all’ when considered in terms of entities [‘it’s’ or things] e.g. numbers, yet really it may not be considered in context to anything other than itself imho.

Regardless of all warpings and loopings physicists dream up

if the universe had but a few stars it would have an edge, if it were but one galaxy - it would have edges, yet somehow because of its vastness it has edges! i can see the idea of curvature in the analogy of a spaceship that keeps going forwards in one direction only to end up where it started [like the world is round]. but what is you are not travelling? are there no planets where people could look up at the night sky and see stars at one side and nothing on the other side?

 

[Postmaster]

I think the mind has an infinite capacity.

 

[_Z_]

yes i agree china cat was right

i think we can 'know' the tao, as we can know infinity. they are simply not a form of knowing where attributes are gathered together to form a meaning - quite the opposite perhaps.

 

[Postmaster]

Sorry you answered to my edited post. My fault.

 

[seattlegal]

Again I say the real challenge is in creating a finite universe..

I'll have to think about that...

...and is nothing something?

...only if you think about it.

 

[China Cat Sunflower]

So, is infinity, for all intents and purposes, purely conceptual?

Chris

 

[Tao_Equus]

So, is infinity, for all intents and purposes, purely conceptual?

Chris

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Ok Ok Ok !! Maybe!!

 

[juantoo3]

Kindest Regards, all!

Quite a thought provoking discussion you have started, z!

here’s the 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?

Not being a mathematician, my first observation is that the puzzle is flawed in its beginning presumptions. I believe others have touched on the issue, but I'll take a stab at it from a philosophical view...

IF the hotel already has an infinite number of "full" rooms, then an infinite number of guests are already accommodated. That is to say, there can be no more "new" guests because they have already been "lodged" before they "arrived."

As for the universe having an end or edge...who are we to say with the limited view we hold? Perhaps the universe is finite, in some sense we might agree...beginning, middle, end, and "size." However, these quantities and measures are simply too vast for us to fathom with the tools we have (our minds and our instruments). I suppose one analogy (from Christian parlance) is the number of hairs on our head, or the number of grains of sand on the seashore. Obviously, these are "numbered," but that number is relatively beyond our grasp. Today, we can calculate into astronomical terms...billions upon trillions upon quadrillions upon quintillions...but these are mostly abstract hypotheticals arrived at by calculating with far smaller numbers, extrapolating the data, and extending the sequence farther afield. Too wit, we may not know "accurately" the full number of hairs on our head, but we can take a representative sample and apply it to a measured map of our cranial anatomy. Same concept can be used to calculate "the sands of the seas." But...if one is striving for absolute factual "truth"...then one must add a "fudge factor." "We calculate that there are X number of grains of sand, plus or minus a certain percentage (fudge factor)."

When it comes to calculating the width, depth and breadth of the universe (assuming "only" three dimensions), there are too many variables yet unknown to us. I feel this is why the universe is typically presented as "infinite." According to Big Bang theory popularized by Dr. Hubble, the universe at one time was an immense mass crowded into a hopelessly small size. I have heard some astronomers suggest as small as a teaspoonful. Can you imagine a universe, with all of the matter and energy known and unknown condensed to such a small size? Yet...at that point in time...what lay beyond the universe? Here is where the question of whether nothing is something may be relevant. For if "nothing" lay beyond that teaspoonful of condensed material, wow! Fast forward...is that same nothing still just beyond the outer limits of the universe? We have no way of knowing at this point in time, but I can see where it leads to all manner of speculation, such as string theory and multiple universes.

Infinity is a concept we use, philosophically speaking, when the concepts are too large to grasp, too large to wrap our minds around. No matter (pardon the pun), we have enough to struggle with just dealing with the astronomically large numbers and sets we know exist and cannot fully comprehend without "educated guesses." I cannot state emphatically that the universe is a sphere 28 billion light years across at the equator...I haven't been to the edge to measure. What's more, by the time we "could" get to the edge to measure, it will have already grown even more! In that dynamic "growing" sense, I think the universe is actually infinite. Of course, then raises the quandary of more and more space filled by an unchanging amount of matter and energy...

My humble two cents.