Infinitely unique

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Infinitely unique

Postby mutestorm » Mon Feb 15, 2010 6:02 am UTC

Let me start this thread by saying I have no college math or science experience and am unfortunately resigned to wikipedia in terms of learning these things. Also, I'm not sure if this should go in math. I probably won't respond too well, or at all, and will only try to follow whatever ideas are generated.

Now, being unable to wrap my head perfectly around infinity (which I hear is a pretty common problem) me and a friend began posing the possibility of an omniscient being who could very well control earth. This is not the topic I'm posting.

What I'm wondering is, if, in an infinite universe, are there infinite types of matter, and/or compositions? If there are infinitely different things, does this allow for something..anything...unique? Is there some inverse probability inherent in infinity that says theres something finite?

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Re: Infinitely unique

Postby rattusprat » Mon Feb 15, 2010 8:22 am UTC

To simply answer your question, I'm pretty sure there is only a finite quantity of matter (or more precisely energy) in the universe. Someone please correct me if I'm wrong.

To take a fairly sharp turn from your initial question: regarding the infinite nature of the universe, I think one one of these 2 options is correct. Could someone who knows better please let me know which (if either) is correrct - or if this is known. Note my knowledge of physics is limited to first year university, with small amounts of additional reading.

(1) The universe wraps around on itself - if you keep travelling in one direction at a constant speed you will eventually come back to the same point (after a very loooong time).
(2) The above doesn't apply, but due to the warping of spacetime from the effects of general relativity there is no "edge" of the universe. In fact, from all points in the universe mass is still (approximately) evenly distributed in all directions. This option is very hard for me to conceptualize.

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Re: Infinitely unique

Postby Zamfir » Mon Feb 15, 2010 10:56 am UTC

For starters, it's perfectly possible to have an "infinity", however defined, without covering alll possibilities. Take the most simple example of something "infinite": a series od sets {1}, {1,2}, {1,2,3}, etc. You can clearly make an inifite amount of such sets, but for example {3,4,7} will not be among those sets.

It might be possible that the universe is really infinite in space, but the assumption there is that all parts of the universe resemble each other statistically (of course, we can never check this). So with the same laws, the same relative amounts of particles, formed into the same stars and gas clouds and galaxies and larger structures. That's at least what the entire observable universe looks like.

In other words, if you take a cubic box of universe and look at average properties of that box, they will be close to the same for the next box. That's clearly not true for a box of 1 meter cubed, or even for a box of galaxy size (one box has a galaxy, the next box doesn't), but for boxes containing billions of galaxies it appears to be true.

Current cosmological theories suggest that the entire universe has its origin in a small (almost point-sized) region of space, at a finite time ago. In which case it can't be truly infinite at this moment, although inflation theories allow that such a finite universe could be incredibly much larger than the part we can even in theory observe. But start-of-the-universe theories are far from settled, and as far as I know there is some room for a truly infinite universe. (Although I think it is pretty much settled that the entire <i>observable</i> universe derives from a small patch of space, and it's not that clear if there is any point in wondering about a possible rest of it)


But let's get back to the boxes. Let's assume that the universe is infinite in space, and that the entire universe can be divided into inifitely many boxes that are on average similar. The question then becomes, must the universe contain all possible boxes? If there is only a finite number of box states, then a universe with an inifite amount of boxes must at least repeat some boxes (and repeat them infinitely often), and if box states are chosen more or less at random, than we would expect every state to reappear somewhere. So are there a finite amount of box states?

A box has finite dimensions and contains a finite number of particles. But a particle's location (or some other property, like its energy) behave as a point on a line: there are more "locations" between 0 and 1 meter than there are natural numbers, so even a single particle in a box has more possible states than there can be boxes in an infinite universe. Which suggests that every box is unique even in an infinite universe.

So the next question: when are two box states "the same" and when are they "different"? Let's say two boxes have the almost same situation, with all particles' locations (or really their wave functions) the same within a millionth of an atomic radius. Does that count as different?

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Re: Infinitely unique

Postby mutestorm » Mon Feb 15, 2010 12:26 pm UTC

Zamfir wrote: A box has finite dimensions and contains a finite number of particles. But a particle's location (or some other property, like its energy) behave as a point on a line: there are more "locations" between 0 and 1 meter than there are natural numbers, so even a single particle in a box has more possible states than there can be boxes in an infinite universe. Which suggests that every box is unique even in an infinite universe.

So the next question: when are two box states "the same" and when are they "different"? Let's say two boxes have the almost same situation, with all particles' locations (or really their wave functions) the same within a millionth of an atomic radius. Does that count as different?


I guess I should clarify as best I can. A pure repeat (by my own definition for the purpose of this topic) occurs when the state of particles in whatever combination repeat themselves. I guess space and time don't count, although I'm tentative to say this has no logical effect on whether something is truly repeated. again, for sake of this little niche of thought, I think the state of the particles must repeat at any one point in the infinitely large (and infinitely enduring?) universe, so they don't have to exist within the same space. Their location within the universe can differ, but their location in making up said box cannot, because it wouldnt truly be the same box if even a single particle had a different mass, dimension, or location on the box.

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Re: Infinitely unique

Postby Zamfir » Mon Feb 15, 2010 1:52 pm UTC

Mutestorm, just to get an estimate of your current knowledge, do you understand the difference between the infinity of the real numbers and of the natural numbers? If not, you might want to look into that, you'll find it very interesting and crucial to even begin to understand a question like this.

If you already know that stuff: a state of a particle in a box is a real number, while the number of boxes in a universe, even an infinite universe, is a natural number. The number of box states is therefore much "larger" than the number of boxes, in the folowing sense:

Suppose you put one particle in a box, with the particle at a random location, and you describe this particle's position with only one number (let's say you add the x, y, and z coordinate).
Now take a new box, and put in a new particle at a random location, describe in one number. And a third box, etc. Even as the number of boxes goes to infinity, the odds that you have two boxes described by the same number will always be zero.

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Re: Infinitely unique

Postby mutestorm » Mon Feb 15, 2010 2:54 pm UTC

Is this coordinate naming system arbitrary? Are the coordinates named in relation to it's would be placement on the box, or from its position in relation to the universe? Also, is the existence of each box not statistically independent? Determinism aside because cause of repetition is not an issue, only that it exists.

And I did not know what a natural number was, but i can still remember real numbers. Is the naming system somehow dependent on the incarnation of the box? If so then it doesn't suit the term for a replica under my definition. I think approaching the definition of an exact copy under the guidelines of the impossibility inherent in counting infinitely isn't constructive really.

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Re: Infinitely unique

Postby Zamfir » Mon Feb 15, 2010 3:47 pm UTC

mutestorm wrote:Are the coordinates named in relation to it's would be placement on the box, or from its position in relation to the universe?

Relative to the box, so two indenitcal boxes would have the same number. The exact details of the scheme don't really matter, just this: if you need one or more real numbers to describe the state in a box, then there will be more possible states than there are natural numbers, and the number of boxes in an infinite universe will not be larger than the number of natural numbers.

Even simpler: there are more real numbers between 0 and 1 (or between any other two real numbers), than there are natural numbers (and there are of course an infinite amount of natural numbers). In fact, there are so many real numbers that if you assigned a real number between 0 and 1 to every natural number, you would still have have covered 0% of the the real numbers between 0 and 1. The infinity of the real numbers is an inherently different, and larger infinity than that of the natural numbers.

mutestorm wrote:I think approaching the definition of an exact copy under the guidelines of the impossibility inherent in counting infinitely isn't constructive really.

I'm not sure what mean here, but keep in mind that "approaching" is really the only way to deal with infinities. You can't deal with them directly, so you have to first describe soemthing interms of finite amounts, then find a rule to see how this changes if you make the amounts larger, and finally try to see what happens if you increase the amount without bound.

For example: I pick a number randomly between 1 and 10. The odds that I picked 5 is 1/10. Now I pick another number, and that odds that at least one of them was 5 is 0.19. I can increase the number of picks, and the odds of picking at least one 5 gets closer and closer to 1, although for no finite number are th odds really 1. But I can get as close to 1 as I want, so we say that "in the limit of inffinite picks, the odds of picking a 5 approach 1". That's basically the only way you can reliably works with infinities.

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Re: Infinitely unique

Postby Tass » Mon Feb 15, 2010 9:02 pm UTC

Zamfir wrote:Current cosmological theories suggest that the entire universe has its origin in a small (almost point-sized) region of space, at a finite time ago. In which case it can't be truly infinite at this moment


That is not true. The universe can indeed be both infinite and expanding. According to general relativity that actually seems to be the case (as the universe appears not to be closed). We can't really know if quantum gravity will change that, but it is not likely.

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Re: Infinitely unique

Postby mutestorm » Tue Feb 16, 2010 3:25 am UTC

If

Zamfir wrote:For example: I pick a number randomly between 1 and 10. The odds that I picked 5 is 1/10. Now I pick another number, and that odds that at least one of them was 5 is 0.19. I can increase the number of picks, and the odds of picking at least one 5 gets closer and closer to 1, although for no finite number are th odds really 1. But I can get as close to 1 as I want, so we say that "in the limit of inffinite picks, the odds of picking a 5 approach 1".


is true then why is

Zamfir wrote:For starters, it's perfectly possible to have an "infinity", however defined, without covering alll possibilities. Take the most simple example of something "infinite": a series od sets {1}, {1,2}, {1,2,3}, etc. You can clearly make an inifite amount of such sets, but for example {3,4,7} will not be among those sets


true?

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Re: Infinitely unique

Postby Cynical Idealist » Tue Feb 16, 2010 6:33 am UTC

mutestorm wrote:If

Zamfir wrote:For example: I pick a number randomly between 1 and 10. The odds that I picked 5 is 1/10. Now I pick another number, and that odds that at least one of them was 5 is 0.19. I can increase the number of picks, and the odds of picking at least one 5 gets closer and closer to 1, although for no finite number are th odds really 1. But I can get as close to 1 as I want, so we say that "in the limit of inffinite picks, the odds of picking a 5 approach 1".


is true then why is

Zamfir wrote:For starters, it's perfectly possible to have an "infinity", however defined, without covering alll possibilities. Take the most simple example of something "infinite": a series od sets {1}, {1,2}, {1,2,3}, etc. You can clearly make an inifite amount of such sets, but for example {3,4,7} will not be among those sets


true?

Because, in the first case, there is a nonzero probability of picking 5. In the second case, the rule for constructing the sets precludes the existence of {3,4,7} in the series, as all of the sets start with 1 and contain sequential natural numbers.
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Re: Infinitely unique

Postby Zamfir » Tue Feb 16, 2010 10:12 am UTC

Don;t get too hang up on the {3,4,7}, it was jsut an example to show that "infinite" doesn't mean "contains everything"

Tass wrote:That is not true. The universe can indeed be both infinite and expanding. According to general relativity that actually seems to be the case (as the universe appears not to be closed). We can't really know if quantum gravity will change that, but it is not likely.

I thought measurements show that the observable universe is flat, or close to it. And that being flat can mean that the universe is spatially open, or that inflation made a closed universe so large that our patch in it is essentially flat. But I am not an expert there at all, so I might be wrong.

I also thought that if the universe is open now, it must have been spatially open since the beginning (or not have a beginning at all). So that even if it is expanding, it was always infinite in space. But again, not an expert here.

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Re: Infinitely unique

Postby thoughtfully » Tue Feb 16, 2010 10:52 am UTC

If you'd like a good introduction into the mathematics of infinity, you can't go wrong with One, Two, Three, Infinity by George Gamow, a noted physicist. You can find an affordable Dover edition at any decent college bookstore, or read it online.
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Re: Infinitely unique

Postby gmalivuk » Tue Feb 16, 2010 6:11 pm UTC

If the {3,4,7} example is confusing, consider a much simpler example:

There are obviously an infinite amount of even numbers. However, they obviously do not contain everything because in particular they leave out all the odd numbers. Or consider a line in space, which is infinitely long but which doesn't even include points that are "very similar" to ones on the line (i.e. points which are arbitrarily close to the line but not on it).

So for an infinite set of [something]s to contain every possible [something], or at least to *probably* contain every possible [something], it needs to meet certain additional criteria. For one thing, the infinity of the infinite set of things must be at least as large as the number of possible things. A countable list can't contain every real number, or every set of natural numbers, or so on. But even if they are the same size (like the number of integers and the number of even integers), it's not always true that an infinite set of one will probably contain every element of the other.

The Champernowne constant and Liouville's constant both have infinite non-repeating decimal expansions, but while the former is guaranteed to eventually include every finite sequence of decimal digits, the latter doesn't even include all the ones containing 0 and 1.
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Re: Infinitely unique

Postby Waylah » Wed Feb 17, 2010 11:03 pm UTC

Tass wrote:
Zamfir wrote:Current cosmological theories suggest that the entire universe has its origin in a small (almost point-sized) region of space, at a finite time ago. In which case it can't be truly infinite at this moment


That is not true. The universe can indeed be both infinite and expanding. According to general relativity that actually seems to be the case (as the universe appears not to be closed). We can't really know if quantum gravity will change that, but it is not likely.


waaait a moment. I thought that although the universe is not closed, doesn't have an edge, it is finite. like the surface of a sphere. I mean, it -used- to be finite, so when did that change? I thought the universe was a finite size, huge, expanding, and has no boundaries.

I'm just talking about the universe at any one point in time, like, RIGHT NOW! :O, as opposed to the universe over all time.

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Re: Infinitely unique

Postby thoughtfully » Wed Feb 17, 2010 11:30 pm UTC

The only conclusion is that if the Universe is infinite, it was always infinite. This doesn't jibe with the intuitive picture of the Big Bang, but I expect it can be consistent with the math. But IANAGR.
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Re: Infinitely unique

Postby RogerMurdock » Wed Feb 17, 2010 11:33 pm UTC

thoughtfully wrote:If you'd like a good introduction into the mathematics of infinity, you can't go wrong with One, Two, Three, Infinity by George Gamow, a noted physicist. You can find an affordable Dover edition at any decent college bookstore, or read it online.


I read the first couple pages online and ordered it off amazon for a few bucks, seems like a sweet book, thanks for the link.

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Re: Infinitely unique

Postby thoughtfully » Wed Feb 17, 2010 11:40 pm UTC

Yeah, they don't make em like good ol' George anymore. He was more than a bit like Feynman. Both were great at making science accessible (I think Gamow might have been one of the best ever), and both were notorious tricksters. Gamow added Hans Bethe to a paper predicting the relative abundances of elements created in the Big Bang just so he could call it the "alpha, beta, gamma" paper, since it was coauthored with his student, Ralph Alpher.
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Re: Infinitely unique

Postby gmalivuk » Thu Feb 18, 2010 1:17 am UTC

Waylah wrote:I thought that although the universe is not closed, doesn't have an edge, it is finite. like the surface of a sphere.

No, a closed universe is the one that's like the surface of a sphere. Ours appears not to be.
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Re: Infinitely unique

Postby Zamfir » Thu Feb 18, 2010 11:20 am UTC

gmalivuk wrote:No, a closed universe is the one that's like the surface of a sphere. Ours appears not to be.

Is there any experimental way to differentiate between an open universe and a closed universe that is much, much larger than the observable part of it?

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Re: Infinitely unique

Postby BlackSails » Thu Feb 18, 2010 1:35 pm UTC

Yup. You use gauss's miracle theorem (or something along those lines) and make a giant triangle and measure the lines. If the triangle is big enough, you can measure the deviation from euclidian space. Sure, if the universe is closed and the curvature small enough, it will look euclidian even at large scales, but that just means you need to measure your triangle better.

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Re: Infinitely unique

Postby Zamfir » Thu Feb 18, 2010 1:46 pm UTC

BlackSails wrote:Yup. You use gauss's miracle theorem (or something along those lines) and make a giant triangle and measure the lines. If the triangle is big enough, you can measure the deviation from euclidian space. Sure, if the universe is closed and the curvature small enough, it will look euclidian even at large scales, but that just means you need to measure your triangle better.

Sure, but I was more thinking along the lines of an inflation-type scenario in which the curvature of the universe becomes really, really close to flat. Is there a point at which you can reject that hypothesis?

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Re: Infinitely unique

Postby gmalivuk » Thu Feb 18, 2010 3:02 pm UTC

Well no, because there will always be at least some experimental uncertainty. So by that method you can only ever say curvature is within some small interval of zero.

Of course, it's possible that there turns out to be some theoretical reason to strongly prefer a curvature that is exactly zero over one that's very close to zero, in which case we'd conclude that the universe is most likely flat. (Analogous to the fact that gravitational effects have only been experimentally determined to travel within a fairly narrow range of the speed of light. Sure, those experiments alone can't tell us that it's exactly c, but since the influences of the other forces travel at c, it makes a lot of sense that gravity would as well.)
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