conceptualizing 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!