conceptualizing infinity

What a perfect forum to share a concept that has been amazing my mind for the past month. Im sure some will find this as intriguing as I do and of course others will not give a patsy.
Im taking this course at school on Physics Theory, and obviously no course on physics theory is thorough without profound class discussion on general relativity.
During a lecture concerned with the topology and size of the universe we got into the contents of finite variance and infinity. Everyone knows what infinity is - neverending and possibly boundless - but like myself previous to this class I think that most people fail to grasp the actual concept of NEVERending. For those like myself who haven't really appreciated the concept of infinity objectively, mull this over in your minds for a bit:

In an infinite universe of finite variance (countable number of particles and atoms) there is no other option but that everything has an exact copy or doppelganger of itself. So there is not only one other exact copy of Franco (me) in the universe but an infinite number of them traversing the cosmos, habitating on an exact replica of this planet, sitting on exactly the same couch, punching keys on the exact same laptop. We may be just copies of the originals, ourselves.

That this is a necessary truth, as long as the two concepts of an infinite universe and finite variance hold true, is quite awe-inspiring.

That a countable number of items can only be put together in so many combinations before the same combinations have to start repeating themselves and that in an infinite medium these combos would be repeated an infinite number of times, really bangs home just how intense a word infinity really is.


Its great to have the perfect forum to vent minded concepts, lol

Well Franco,

The problem as I see it is the concept of "finite variance"..............why should it even be countable, and, even if it were, why does that preclude the possibility of everything being different?

OK we know that at atomic and sub-atomic levels there is a fair degree of replication, but as soon as you get to meso and macro levels, this must be far less probable?.....................basically there are far more combinations and permutations of the lower level "building blocks".

Also, the Universe may be infinite, but it isn't full, and there is no reason that I can think of why it should be? You would only get the first true replication as you seem to be describing it, if all the possible combinations and permutations had been exhausted and the system was continuing to grow.

Similarly, there is no particular reason why exact replicas should not exist anyway, by chance? After all we know that they exist at the lower levels.

I guess I am arguing that it is more to do with the laws of probability than the concept and mathematics of infinity:)

I notice that I didn't explicitly state that we have to assume both finite variance and spatial infinity have to be true for this to be a result.

Fair comment you made on whether space is fill or not - I want to respond to it further but its a friday house and my night has guests, lol


there's an interesting article in the April '99 issue of Scientific America titled 'Is Space Finite?' - I'd upload it but Im not sure of the legal issues since Iv paid for it.

You can try here but im not sure if it will work -
http://www.sciamdigital.com/index.cf...9C-94564BA4BE9

It sort of addresses questions of bounded or unbounded infinity . . .

most def the laws of probability apply - pi, the quantification of infinity - microcosmically, if written out to about 1500 places, one would be able, by the laws of prob, find any and all combinations of 3 digit combinations. So if pi were written out infinitely then any combination of any amount of digits could be located within pi an infinite amount of times.

if your interested in that article IM me ...

The thing that I always consider when trying to think of infinity, is how large the insides of infinity are: i.e. the number of sub-atomic molecules, to make and atom, to make an element, compounds and so on. (Approximately) if you think of every letter of every book in basically all the world, you are thinking of the same number of letters as cells in a human body... it makes you think.

Hi Franco, have a good week-end:)

I think We need to define some terms a bit here? me being from the UK and all?

By "Universe" I mean the whole thing, or "Space", which is supposedly infinite. I would be inclined to use the expression "Cosmos" for the part of the Universe that is occupied, or partially occupied at least.

Now, we would assume (in theory at least) that the Cosmos is measurable and finite, but the environment it is operating in is not. For all we know there could be numerous "Cosmoses" which we would be totally incapable of detecting?

"Bounded Infinity"....................that sort of reminds me of Euler (spelling) and his "Aleph zero"???????????????

I personally think that a "bounded infinity" is a contradiction in terms. It is rather "lawerish" but it puts me in mind of our property laws here. I own my house freehold, which means permanently (infinitely?). In the past I had a place with a 999 year lease.............. a "virtual freehold" so to speak. the first title is infinite but the second is finite but irrelevant, as it is effectively infinite.

I don't agree with your comment on Pi though, as if you wrote it out "infinitely" (assuming it is infinite) you would never finish, so you can never be sure as to how many combinations there were.

I am inclined to take the view that the laws of probability do not work well on a population that is infinite, or at least become highly questionable. The reliability of statistics declines with the size of the sample population being small and the total population being large? Well, infinity is a pretty large population:D

Johnno

I am trying to get my head around something which I have been thinking about (and it may sound silly, but I am 15 give me a chance :D):
if we take the universe being of an infinite size to be true, and if the universe is always expanding, then isn't infinity become greater than infinity, therefore infinity is increasing... but it isn't... if you understand that. :D

-Chris-, I am not a great expert here, but It might help if you sort of mentally separated spatial infinity (astronomy or whatever) from mathematical infinity (number theory?).

Multiply 500 by infinity and the result is infinity. Multiply infinity by infinity and the result is infinity. Divide anything other than zero by zero and the result is infinity:D ( the famous divide by zero error in programming)

Space, or the Universe as I call it, is not expanding............the cosmos (the occupied bit) is expanding. Another way of trying to rationalise it is that if something has no end, it doesn't have a starting point to expand from?

Perhaps my posts should have some sort of government (mental) health warning? :D

Haya Nihil, Chris - thanx and weekend tidings to you both

Ive been coming back and forth trying to stay with the thread so my posts are getting shorter, lol

an example of bounded infinity would be the type of universe (or cosmos - I love that word) in which segments of time space are juxtoposed consecutively in all dimensions and an 'asteroid game' like medium - or one in which a space traveller flying to the right would exit to the right and immediately reappear from the left.

there are other examples of bounded infinitys as well ...

Hmmmm, "donut theory"?................and then we have "dimension X" ?

:)

the paradox of an expanding infinity is a good point - there are postulates - and Im not specifically speaking for myself here - that the galaxies or any cosmic matter are not actually moving farther apart, but rather space is accumulating between them - that may help conceptualize expanding infinity - its not so much journeying outward from its edges, in this theory, but being shoved outwards by its infrastructure.

Theres another flabbergasting concept along these lines - that all points are at the center of infinity ...

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that all points are at the center of infinity

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Not that hard to understand at all really. So if we take it to be true that infinity=infinity, and that if you can name any number there is always an infinite amount of numbers higher than that number. and lower than that number. Just apply that number to mean some distance of space. So If I'm at a point of space, lets call it 500, there is an infinite amount of space traveling out in any direction. Now go in the exact opposite and you are once again measuring an infinite amount of space. So if you can always measure an infinite amount of space in any direction, you are in the center...

I believe Galileo was one of the first people to conceptualize this paradox about infinity.

Hmmm, or maybe the centre is zero (nought) and you are always there because you have plus infinity and minus infinity ahead and behind you respectively?

:)

Wow. So, if everything has infinity around it, everything is the centre of infinity?

Hi

Interesting discussion :) Let me add a few thoughts.

By the way: beautiful explanation, mohaughn, about the center of infinity.

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So, if everything has infinity around it, everything is the centre of infinity?

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Depends on your definition of centre and the metric you apply. Everyone knows what
the centre of a circle is (point with the same distance to all circle-"elements") -
but how do you define the centre of a square (although everybody knows what the centre
of a square is, right?)...
but in simple words: yes



Back to the original poster:

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In an infinite universe of finite variance (countable number of particles and atoms)
there is no other option but that everything has an exact copy or doppelganger of itself.

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This is true, however, its formulation is a bit ill-defined, because particles and atoms
can have different "states". See my definitions below - thus the variance I use is not the
same variance mentioned here.



Before I start some arguments, I want to show something mind-boggling, in order to
illustrate the difficulty we face here.

Simple question: How often do I have to throw a dice in order to be 100% sure to throw
at least once a "1".
Simple answer: 6 times....well, that's wrong.

Correct answer: infinitely many times.

The probability in one throw not to hit the "1" is 5/6. The probability in two throws
not to hit the "1" is (5/6)^2 = 25/36,
and in n-throws (5/6)^n.
Thus, the probability to at least hit the "1" once in n-throws is 1-(5/6)^n. This probability
now should be 100% = 1. Hence, in order to be 100% sure to hit the "1" at least once, one has
to throw

n = log(1-1)/log(5/6)=log(0)/log(5/6)= infinity

times....interesting, isn't it?



I tricked. Of course, because the question we face here is a different. There are two important
ingredients to consider. First, does our system have a memory, second, is our system countable
infinite, or uncountable infinite.

Remark: Human beings assume when playing Roulette that the wheel has a memory. It doesn't!!




definitions


Let me allow to draw the following picture:

- an "item" is represented by a dice with n-faces.
E.g. n=6, the dice has 6 faces with labels 0,1,2,3,4,5. This is my "variance".
If n < infinity, then the variance is finite, if n is infinite, then the variance is
infinite (more precise: countable infinite).

- the "space" is completely filled with the dices, thus we have a "memory" (as opposed to
the above calculation). If the number of dices is finite, space is finite, if the number of
dices is countable finite, then space is countable finite, if the number of dices is uncountable
finite, then space is uncountable finite.

- a "cluster" is a connected set of particular items. A finite set of items in a cluster makes
it a finite cluster etc.




finite variance and finite space


Let's fix the ideas to n=6 (a dice). All calculations are feasible, because variance and space are finite.
Example: 100% probability to have at least one replicant: space must be of size 7 (7 dices).

Certainly, one can calculate the size of "space" needed to find at least two clusters consisting of two "1",
three "1" etc. with 100% probability....so, what is the size of needed "space"? Right, countable infinite!

For example you can calculate the "space" needed to find two Francos...just
do not forget to incorporate that an human-being friendly environment also is
needed. Hence, in order to find two Francos, you need to find aan earth-like
planet in a solar-system like environment...again, what is the size of needed "space"? Right, countable infinite.



infinite variance and finite space


It becomes trickier as you may imagine, since the probability to actually throw a "1" (remember: a memory is present)
actually is lim(n -> infinity) 1/n = 0.

Thus, mathematicians assume for their calculations that n is finite - perform their calculations, and then let n go to infinity.
Does that make sense? yes, usually, it does. A simple example:
- n -> infinity
- n!=n*(n-1)*(n-2)*... -> infinity.
- However, n/n! -> 0 for n-> infinity. Why?
n!=n^2*(1-1/n)*(n-2)*...
n/n! = n/(n^2*(1-1/n)*(n-2)*...) = 1/(n*(1-1/n)*(n-2)*...) -> 0 for n -> infinity

In this way, the probabilistic calculations can be done. Anyway, while some numbers must be there, the
probability to find a replicant if the variance is infinite is zero.



finite variance and infinite space


While it is intuitively true that every cluster has a replica of itself, it actually is only true depending on certain
circumstances:

if the cluster-size is finite, then this is correct. I guess this is what the thread-starter meant.

if the cluster-size is countable infinite, then this is only correct of space is uncountable infinite.
if the cluster-size is uncountable infinite, then I have no idea.


An example to illustrate this point:
- finite variance: 0,1,2,3,4,5,6,7,8,9
- an (ordered) cluster (of size 3) consisting of 2, 3 and 7 can be represented as a real number |R: 2.37.
Thus, the set of real numbers can represent clusters of any size.
- countable infinite space can be represented as integer number |N_0

-> Since the set of integer numbers |N_= is a null-set in the set of real numbers |R, you cannot guarantee
for replicas!
-> However, how do you define a "replica". Here, I was extremely restrictive by demanding ordered cluster.
Combinatorics allows for additional factor, which may change the result enormously (for simplicity, I
have not mentioned it above).



infinite variance and infinite space


This gets more and more complicated, but you have to throw in the limes-technique described above (probabilistic),
combinatorics, plus the discussion about countable/uncountable infinite sets. I guess, myself I would now spend
hours over hours to work it out - for the particular cases.



causality


Just for completeness, because relativiy has been mentioned: there is an another,
very interesting aspect to consider - the aspect of information flow (also an intrinsic part
of relativity theory, sometimes refered to as causality).
Information cannot propagate faster than light, thus all information we can gather
must be part of the backward light cone. For us, in our reference system, you may
say that there is a non-negligible possibility of having observed identical replicas
at the same time in our past...




Cheers :)

I just use "The Force"

Its easier that way.

public bool takeMeToInfinity() {
takeMeToInfinity();
return true; // Ta-da!
}

(thanks to last-op-return optimization, this won't eat your memory :)

Sorry, I didn't mean that the concept of all points in an infinite medium are its center was incomprehensible, I just meant that it is mind boggling in its simplicity because its not a concept that is immediately grasped or intuitive without some encouragement. lol

Mo's example using a number line to help visualize infinite space and omni-point centricity can also be used to conceptualize an 'expanding infinity' bought up before.

I believe it was previously bought up that it has been theorized that space is not actually expanding, as in galaxies moving farther from one another, but space develops between them - so bits of space are added between galaxies causing expansion rather than objects in space actually moving apart from one another.

If we take Mo's number line we can see this -


...-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 ...

now if we add, say, .5 to all numbers we have -10.5 -10 -9.5 -9 -8.5 -8 ....... etc.

from -10 to 0 is still, quantitively, the same distance - there are just more steps to traverse to get from one to the other. This illustrates well the theory that space is not actually expanding but chunks of space are added between cosmics entities pushing them away without these cosmic entities really moving farther distances away.

The other thing that can boggle the mind when you are thinking about the ability to always have an ever smaller amount of space between things is that no two things can actually touch. Given that there is an infinite amount of space between things as you can always define a smaller measurement there is always some amount of space between things...

Gets even more interesting when you use this same logic and use it to talk about something falling into a black hole. At the center of the blackhole is the event horizon, and you will continue to fall closer and closer to the event horizon for eternity as you can only get close to it, but never actually reach it, you just get infinity close... Of course a lot of this also assumes that you can't travel at the speed of light and is being really over simplified.. I think the latest theories on black holes is that they convert matter into energy and as energy does travel at the speed of light you will eventually hit the event horizon, but you are not what you were..


And then to confuse things even more there are some recent theories that the universe is not infinite and is not expanding, but is rather like a hall of mirrors. Such that when you look deep enough into the universe in any direction to see the edge of the universe you are actually seeing a reflection of some other section of the universe. And if you could peer deep enough into that reflection you would again start to view another reflection, thus giving the perception that the universe is infinite.

Or that the universe is not infinite but rather shaped in like a ball, some say a soccer, others a football. Think about starting at any point on a football and just start drawing a straight line on the ball, as you move away from the original point you start to fall which allows you to continue moving in any direction for an infinite distance. Eventually if you draw a long enough line, you end up where you started... We already know that space and time can be curved, and we have hard evidence to prove that gravity can bend space and time. I tend to believe that the universe is curved and what we perceive to be infinite is something far different that we currently can't comprehend...

I tend to agree with Kant that what we see and perceive is limited by our own mind and understanding. We are fundamentally limited in what we can understand because the proverbial wool has been pulled over our eyes. We perceive what we perceive in the phenomenal world because it is what seems orderly and proper to us, but we are the ones that decided what is proper and orderly. The noumenal world could be vastly different but we are incapable of perceiving it for what it really is.

This all becomes very clear when you start looking at the outcomes of string theory. Human minds can only spatially realize 3 dimensions, 4 if you include time. But we have a hard time really getting our heads around time as a dimension, atleast something that can be traversed in different directions like the other three dimensions. With string theory you start to get into a universe with many more dimensions. It starts to be something that you can only understand mathematically and not intuitively.

Such a great topic and yet no mention of chaos physics :mad:

If you incorporate the butterfly effect into this whole debate, I think then the interaction of particles (not so much the particles themselves) becomes important.
Identical particles, in identical situations may not ever yield the same results if the interaction between them changes the slightest amount.

Let's throw in the Lorenz Attractor while we're at it.
Something appearing to have a distinct pattern but quite possibly, never repeats itself. Examples, flag fluttering, cigarette smoke, etc. Now you have the notion that even if these particles appear to repeat themselves, there still may exist, subtle differences.

Just a few more thoughts to destroy whatever brain cells the beer didn't kill. :eek:

Haha, I knew that this would get into chaos theory and string theory sooner or later, Im surprised it hasn't gone more quantum yet, lol

The illusion, reflection and mirror theory could be another example of a bounded infinity, its been categorized as that in some theorems - it's not my belief, but minds greater than mine disagree with me.

The article a mentioned earlier in this thread deals with exactly those theories.

I actually sway towards the idea of a curved universe, too. Im not sure if its gravity, rather than mass, that actually curves space though, black holes after all are really just vast curvatures in the spacetime medium.

In my case, it's only been very recently that I have started to be able really grasp time as a dimension, and a spacetime unity. I was involved in a discussion on quantum reality and the person I was with mentioned that even a vacuum has spacetime in it. I can't quite explain why that impacted me so well that I could finally unify space and time and see time as a dimension within itself better. Just that - time is existence (period).

Here's a baffler <chuckle = "sadistically">No 2 objects can occupy the same space simultaneously but 1 object can occupy 2 different locations at the same period in time (or appear to.)</chuckle>
I've never really understood the explanations as to why entanglement isn't evidence of information moving faster than light. If that is so isn't it also so that an electron particle is occupying 2 locations simultaneously?

If that doesn't melt your eyes out of their sockets ...... Einstein didn't even like it, lol!

Hi

I like the discussion here, however, I am a bit short of time so I will
pick out 2 points only.

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No 2 objects can occupy the same space simultaneously but 1 object can occupy 2 different locations at the same period in time (or appear to.)

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That's a strange remark. What's an object? Myself, I am an object, and
at the same time, a part of myself is at the floor while another is up in the air
:)
That's a simple pictures, but actually shows some thruth:

Define object. Then, define how you describe the object: Are you using
wave-function for the quantum object? Are you using a quantum field? Note
that Quantum Field Theory is more fundamental since Quantum Theory, since
the quanta emerge from the field.

In this qm-sense the object actually is everywhere (based on the Kopenhagener
interpretation of the wave function \psi: propabililty |\psi|^2), however,
when you measure it, it is somewhere with a particular uncertainty (Heisenberg).
Since you don't know where it is (exactly), how can you say it is at several
positions at the same time?

Even more - it is still not clear whether electrons are point-like. It is still not
clear how quarks and gluons actually "look" like - the more energy you pull to
be able to resolve space more clearly, the more you see that quarks and
gluons consist of quarks and gluons...

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I've never really understood the explanations as to why entanglement isn't evidence of information moving faster than light. If that is so isn't it also so that an electron particle is occupying 2 locations simultaneously?

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In simple words: because you don't know of entanglement until you compare
results. You only can compare results without violating causality. I have written
something more here[1]. As a consequence of the famous experiment,
quantum cryptology[2] gets its foundation - and _not_ violation of causality.

Cheers :)

[1] http://antionline.com/showpost.php?p...9&postcount=10
[2] http://www.idquantique.com/ (see papers herein)

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If that doesn't melt your eyes out of their sockets ...... Einstein didn't even like it, lol!

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Albert hated the idea. The uncertainty principle underlies the idea of entanglement. And it was tough for Einstein to accept that a local system could affect a remote system. Especially when it happens across the galaxy. But I think even he succumbed to it's logic as Earth time progressed. Reminds my of a favourite quote:

"The more success the quantum mechanics has, the sillier it looks." - A. Einstein.

Einstein was alive with Born Schrodinger and Heisenberg etc and worked with some of them.

Too add a pun or twist on the original post. If a measurement device was "infinitely" precise. There would be no uncertainty in measurement of momentum and position. ;) And it is easy to get caught up with another mirror copy of myself enjoying a hot cup of coffeed floating around time and space in an infinite reality, however we are talking about single "particle" that is a weeeeee bit tiny. If all the particles (million of billions) that make up me ended up at the exact same "time" in a particular "position" and at the same "momentum" that would be highly... impobable. Perhaps infinitely improbable. :)

We could all jump on the Heart of Gold and swish around the universe. (referece to the improbability drive space ship stolen by Zaphod)

Heh, or the bistromatic. That'd be even better. (Sorry for interrupting a serious discussion)

Cheers,
cgkanchi

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No 2 objects can occupy the same space simultaneously...

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I'm a bit rusty on my multidimensional theory, but isn't it possible for 2 objects to occupy the same space provided their molecules vibrate at the same frequency? Please correct me if I'm wrong.

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I'm a bit rusty on my multidimensional theory, but isn't it possible for 2 objects to occupy the same space provided their molecules vibrate at the same frequency? Please correct me if I'm wrong.

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Not directed at me but some thoughts;

It seems that if two objects are vibrating at the same frequency they would cancel one another. It would have be different frequencies?

But in the case of mechanics, two objects cannot be in the same quantim state at the same time. And thus two objects cannot occupy the same "space".

This deals only with matter. Speaking of frequency and my opening statement. Light and radiation can pass thru matter. The key is Thru. But looking at particles in the quantum or classical sense; no two object can occupy the same space. That is have the same quantum state or "spin".

At the molecular level this is especially true. In a popular sense regarding the ideal of alternate dimensions; one could tune to the "frequency" of the dimension they cannot see. Those dimensions still exist within our physical universe. They are just undetectable by our senses. An example could be molecules moving faster than we can detect with out eyes. We cannot "detect" light moving through space as we are, but we can see its effect on objects.

For instance the light reflected from this table. Which happens to be in the wavelength or "frequency" our eyes detect via the retina and the receptors there as "white". Although we cannot see it moving, nor does it occupy space in the sense matter does, as it travels from fluorescent tube to the table.

Objects need to be at opposing frequencies to cancel one another...

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Originally Posted by RoadClosed

Not directed at me but some thoughts;
But in the case of mechanics, two objects cannot be in the same quantim state at the same time. And thus two objects cannot occupy the same "space".
(...)
That is have the same quantum state or "spin"

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Well, this is only true for fermions (simplified spoken). An unlimited number of
bosons can share the same quantum state - which is even visible on a
macroscopic level - the boson-einstein condensate. Superconductivity goes
along the same line, however here, fermions are involved. The (still not
proven, but generally accepted) BCS-theory says that fermions build bosonic
structures (cooper pairs), which then can condensate, ie. they will share the
same (lowest-energy) quantum state.


as per "molecules". Molecules, consists of atoms, themselves consisting of
tiny electrons and a nucleus, which itself consists of protons and neutrons,
themselves consisting of quarks and gluons - essentially, a "molecule" is empty.

The only reason a "molecule" cannot "occupy the space" of another "molecule"
is the interaction of their components -
at large length scale (low energy), this is the electromagnetic force,
at small length scale (high energy) this is the strong force (essentially).
If you accelerate molecules (ie you pump energy in the system), they
eventually will "occupy the same space" when colliding :)



Cheers

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BCS-theory says that fermions build bosonic
structures (cooper pairs), which then can condensate, ie. they will share the
same (lowest-energy) quantum state.

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I read a great deal of the theory when it was tested in Boulder. Kind of forgot about that. I need to look at it again. It creates a "special" material by super cooling paricles using a laser. Back to the physics lab of Colodado State. I thought it created a sort of mass but they didn't occupy the same space. In my argument however I used "quantum state". I shall return. ;)


//EDIT oh and I was refering to fermions.

... well my ****ing head hurts now. Don't have much time but..

;)


As usual when I go back and read somthing, and given the reading of other articles in the mean time I learn something new.

bosnian particles are special. The condensate uses bosnian particles only. Most likely because they use a photon? There is nothing in quantum mechanics that says this cannot happen, in fact it predicts it. This is something new. It is a new form of matter, and the word new is key here. **** when I look at it more I get freaked out. Due to the nature of this substance it could OVERCOME gravity! In the years that have passed more has been made and it's getting easier to store. Can't wait to see what it's used for in the future say... 20 years.

Hi

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Due to the nature of this substance it could OVERCOME gravity!

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Interesting thought. How so? (Are you refering to the floating magnet?[1] :))

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In the years that have passed more has been made and it's getting easier to store. Can't wait to see what it's used for in the future say... 20 years.

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Right :)

Cheers

[1] http://www.youtube.com/watch?v=pO2eDJBr50E

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Interesting thought. How so? (Are you refering to the floating magnet?

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kind of, that is one of the coolest things I have seen. Love the spin. I'll have to read up on how it was accomplished. But super conductors are condensates as well, I think. Here is my thought and it may be way over simplified.

The condensate is at a condition where the particles are in their lowest possible energy state.

Therefore according to this quantum model (Bose?) they cannot give up any more energy. They cannot reduce their energy, cannot step any further.

Not even friction can give them energy. Thus friction ceases to exist. Kind of like in a superconductor?

Without friction there is no way for the particles to adhere to their surroundings. In this case a container.

Thus they will disperse to all areas of the container or just.... "jump" out.