If the universe is infinite, does that mean that everything exists somewhere?

[Sage Lee]

Sage, you may be mixing up two concepts...that of a Line, and that of a Line Segment. By it's very nature, a Line (in the strict geometric sense) is infinite in length. A Line Segment is bounded, and of finite length.

I don't think I mixed these two up, rather, I was arguing that an infinite line running east to west is similarly bounded, albeit in a different fashion, in that it can't ever bend or travel north or south, up or down.

But suprisingly, I (think that I) actually get most of what you told me in this post, on my second read through. You're basically saying (I probably think about this weird, but I think the conclusions are the same) that a line segment and a line, though one might be smaller than the other, are both infinite in the sense that you can theoretically zoom in enough (for lack of a better way to say that) on any given section of pretty much anything, and plot an infinite number of points.

And in this sense, it also seems like you just told me that infinity is contained within finite things, you just have to be capable of going smaller and smaller. Yikes. You're crazy, man. I like you, but you're crazy. (No, just kidding.)

For those unfamiliar with Set Theory, it comes as a real shock to learn that there are EXACTLY the same number of points on the line interval from [0,1] as there are on an interval twice as long [0,2].

I guess I could say I was familiar with Set Theory, since it drives me batgarbage crazy. My stumbling upon Set Theory is actually directly responsible for my attempt at Masturbational Existentiality; I still remember the first thought that I had when I read about Set Theory, it was something along the lines of, "Holy ****, you can talk about anything as math, even abstract or intangible things!" (This may not really be true, but at the time it got me thinking about a supremely omnipotent observer, and what he might or might not be able to observe, and how to quantify all of what he could possibly observe (including intangibles.) It was my discovery of Set Theory that started that whole train wreck line of thinking.

So, to clarify, is it possible to talk about sets containing abstract things, like "the set of all thoughts about hot dogs," or can you only have a set containing objects? Some of the things I said when talking about M.E. a few posts ago are probably even more ridiculous than I realized, as I thought at the time that such intangibles could already be quantified using Set Theory... but now I'm realizing this might not be true. Mehhh, but I so want it to be true!

Perhaps the single biggest surprise, when first learning transfinite Set theory, is that not all infinite Sets are equal.

This was actually not that hard for me to stomach, as it seems to make perfect sense once explained correctly.

For me, the biggest surprise was that an empty set has a cardinality of 1. (Did I say this right?) This just pissed me off, and got me reading about vacuous truth, and it wasn't long before I threw my hands up in exasperation and stopped trying to understand why.

But because of my frustration, I didn't like the joke "in a set of zero mathematicians, anyone of them can do it [change a light bulb]." I actually remarked, to no one in particular, that "in a set of zero mathematicians, three of them are actually tomatoes." I liked this better because, "Hey, if we're being ridiculous, let's just let it all hang out and be ridiculous." What can I say, I was annoyed and was on that previously described tomato kick at the time.

But whatever, I accept on faith alone that an empty set is actually "one," because Wikipedia told me so... but I don't have to like it.

(You have to keep in mind that I and my unschooled mind tried to take in a LOT of very complex information all at once, pretty much on a whim (damn this insatiable curiosity I have to understand,) and for this reason, it's very hard for me to retain much of it. Also because it's not like I ever put any of it into practice, I just thought about it for a while. This was all about five years ago; I don't really remember exactly why I had such a problem with the empty set, or why I said those things I said, I just remember saying them.)

But all in all, I really, really like Set Theory, because as I said, with it, it seems possible to describe just about anything at all using math.

Okay, I just have to share the other joke I came up with when I first read about Set Theory. Alright, ready?

N > Stephen Hawking

I find this funny, but only because I know what N is. In all honesty, I should probably just leave it at that, because if I tell you what N is you'll just think I'm an *******. And besides, nothing is as funny if you have to explain it.

But *sigh* I started it, so I'll finish it: N is the set of all things that can change a light bulb.

Now, to be clear, I don't mean this in any spiteful kind of way. Obviously I can't relate to being in a wheelchair, and I certainly don't understand how it might feel to have that poked fun of, but I really don't mean to be malicious with that joke. I don't intend to slight the man himself in any way; in fact, I'm quite convinced that he can probably shoot laser beams out of his eyes and crumble my very existence with a single, profound thought. Hell, who needs to change light bulbs when you can power them forever with your mind? Rather, I'm poking fun of the absurdity of such a brilliant and existence-crumbling-mind being (probably) unable to accomplish such a simple task (without assistance), one that much simpler folk like myself take for granted.

Forgive me, but I pretty much find everything funny given the right delivery or moment. I'd like to think that if Hawking heard that joke, he'd be wise enough to be able to take it in the spirit it's meant, and to maybe even also find it funny. I don't know, does anyone else find my joke funny, or should I just keep things like that to myself?

Regardless, I still think that would make a great T-shirt (just the joke, without the explanation of what N is.) Visually, to non-math people, it reads "N is greater than Stephen Hawking" (rather than N contains Stephen Hawking) and at it's core is saying, in a roundabout way, that "a light bulb is greater than Stephen Hawking." Frankly, I just find the thought of ANYTHING being greater than Stephen Hawking to be kind of funny, who cares what N actually is?! I would wear the **** out of that shirt, and if anyone asked me what it meant, I'd probably just smile and shake my head. (I'm also aware that "N" in math might already mean something specific, but if you can choose whatever letter you want to designate some set you just pondered, then I choose N, as it's better visually for me than A or B or X or Y or Z. Don't ask me why; I'm particular about these things.)

In my final defense, I'll just point out that I don't find this hilarious or anything, it just makes me smile.

In order to understand this, you need to understand the formal, logical distinction between what is a "necessary" condition, and what is a "necessary AND sufficient" condition. They are not the same. I guess the best way to explain is through an analogy and example.
To say that X is a necessary condition for Y is to say that it is impossible to have Y without X. In other words, the absence of X guarantees the absence of Y.
Example: Having four sides is a Necessary condition for being a Square.
Notice, however, it is not a Sufficient condition. For example, a Rectangle has four sides, as does a Rhombus, but they are not necessarily Squares. A Rectangle has four equal angles, but may not have four equal sides. Conversely, a Rhombus has four equal sides, but may not have four equal angles.
Compare/contrast the above example to the following:
A quadrilateral with four equal sides and four equal angles is a both Necessary and Sufficient condition for being a Square.
-Or- another way of phrasing this: A quadrilateral that is BOTH a Rectangle AND a Rhobus is a Necessary and Sufficient Condition for being a Square.

Out of curiosity, is it correct to capitalize all those words when using math-speak? It never would've occurred to me that it's proper to capitalize Rectangle, but since you took the time to do it in several instances, now I'm thinking it's probably the norm. I find that interesting. As you may have realized, I write a lot, but I don't recall ever having cause to write the word Rectangle.

Now, getting back to your question as to how an Infinite Universe isn't a "Sufficient" condition for "Everything existing somewhere"...
It is a Necessary condition that the Universe be Infinite in order for there to exist the possibility that "everything exists somewhere". This is obviously trivially true, because if it were not infinite, then it would be finite, and a finite Universe cannot be a Necessary condition for everything existing somewhere. So, as a minimum, it is a Necessary condition that the Universe be Infinite in order for this possibility to exist. However, that is not a Sufficient condition. As discussed in earlier posts in this thread, the Universe may be "countably" infinite...that is to say, having the same size (Cardinality) as the countably Infinite Set of Natural Numbers ( |N| ). However, the Set of all Even numbers is just as big (i.e. the same size) as the Set of all Natural Numbers, yet the former Set is missing an infinite amount of numbers...that is, the Odd numbers. So, these two sets have exactly the same NUMBER of elements (members), but these two sets are not "identical", and only one of these sets "exhaust" all the Natural numbers, whereas the other set does not.

I think infinity just doesn't mean what I thought it did at the start of all this. It's still kind of bothersome that something can be infinite and yet be missing an infinite amount of things, but I think I get it now.

With that said, I am not exactly certain what would be both a Necessary and Sufficient condition for an infinite Universe to ensure that "everything exists somewhere". From a purely mathematical perspective, I might argue that the Universe would need to have the Cardinality of the Continuum

Wow, that sounds really cool. If I had to name a band, or an album or something, right now, I'd name it that. It sounds so damn epic.

(= the Set of Real numbers). However, one could equally argue that that, in and of itself, may not even be a Sufficient condition. The tiny interval [0,1] on the Real number line is everywhere Dense and Continuous, and this segment contains an equal number of points as in the entire Real Number line. In fact, it contains in equal number of points as on a plane. Moreover, it contains just as many points as on any finite n-dimensional space. Nevertheless, despite the equipollence of the interval [0,1] with the entire Real Number line, it is not "exhaustive". It doesn't contain the number "2", or "pi", or "e", or for that matter any Real number greater then one or less then zero.

I don't *quite* get what you mean by "dense" here, although I think you're just reiterating what you've already explained in a slightly different way.

All this gobbledygook ultimately comes down to the conclusion that, even though the Universe may be infinite, it does not necessarily follow that "everything exists somewhere".

Congratulations, to both you and Chalnoth. I now completely agree with that statement. Gold star for youse guys. Although I'm thinking, as I said before, that I never really disagreed, I just didn't understand what infinity actually meant (I thought it literally meant "exhaustive.")

 

[Sage Lee]

It doesn't stop, though. It keeps repeating over and over again, endlessly.

Yeah, but it's fiiiiniiiite! <stamping foot and holding breath>

In a real-world scenario, this would be like there being an observable universe somewhere far away that is absolutely identical to our own. If the universe is infinite, in fact, we know this must be the case, because due to quantum mechanics there are only a finite number of possible configurations. So if it is infinite in space, then the real universe would actually behave very much like the repeating number line, except that the repetition would be more chaotic than orderly.

Wow, I didn't even consider this implication. How very interesting.

From quantum mechanics we find that the total number of possible configurations of a given region of the universe is finite. It's a very large number, but a finite one nonetheless.

We are on the same page...

Well, no, because in this case a triangle is an abstract mathematical construct. It isn't a real object. Because it is an abstract mathematical construct, with a very specific definition, we can say absolutely that it doesn't have four sides.

This seems like a funny thing to say. If we were talking about a real object, wouldn't it have an even more (or at least an equally) specific definiton, and couldn't we also say absolutely that it doesn't have four sides? I mean, having something to look at and touch and feel seems like it would be more definitive than just thinking about an abstract concept, so I find it weird that you started that sentence with "because it's not real" Of course, I don't deal much in math, so that's probably why that seems that way to me.

But I get it, I can't really argue with anything you've said on this subject, even if I like to try and play Devil's Advocate.

What you're saying is, you don't believe in magic. (No, don't respond to that, I'm just playing now, and besides, dead horses start to smell after a while, so I'll just sweep this one under the rug and move on...)

No, that's not the right way to look at things. Our choice of logic is more or less arbitrary. One choice of logic is no more or less correct than another. But one choice may be more useful than another under certain situations.

This, though I don't really get. I should probably brush up on my logic. By which I mean to say I need to go back to 101. I never got beyond "if," "then," and the occasional "but," plus I skipped that class all the time and ultimately dropped out because I preferred to smoke weed and play hackisack before lunch. (Oh, who am I kidding, I skipped all my classes, no matter the time of day. In my defense, it was some pretty good weed, and I was a ****in hackisack god. Fortunately, I squeaked by in a few classes because, believe it or not, I'm quite charming in person, and my teachers have mostly seemed to like me. I've been told I have charisma, whatever that means. If it was a girl who said it, it would probably mean "I'm ugly," but fortunately, it wasn't.) Aaaaaanyway, I thought logic was logic, and as you stated before, is consistent. What are these choices of which you speak, and how is that choice arbitrary? Do you care to provide any simple examples of what you just said?

No biggie, if not; after reading that other thread I linked in an earlier post, the one about infinite-infiinte, I couldn't help but wonder how often some new guy comes in here and just up and barfs all over the forum, leaving you guys to clean up the mess, and I appreciate the patience you must have when dealing with people like me. So I understand if not, and I can probably find some good places to learn on the internet, but it's always nice to be able to ask questions and further refine one's knowledge.

 

[Chalnoth]

This seems like a funny thing to say. If we were talking about a real object, wouldn't it have an even more (or at least an equally) specific definiton, and couldn't we also say absolutely that it doesn't have four sides?

Any time we apply mathematics to reality, we have to consider that we don't actually know for certain whether or not the mathematics applies.

In this situation, for instance, there's no such thing as a triangle in reality. You can draw something that looks like a triangle on a piece of paper with a pencil, for instance, but what it really is is a bunch of graphite and rubber atoms spread across the surface of the paper. It simply isn't possible to make atoms form a line segment, because the atoms are of finite size.

Because of this, it is very possible to draw something that looks like a triangle on paper, but doesn't actually have all of its properties.

This, though I don't really get. I should probably brush up on my logic. By which I mean to say I need to go back to 101. I never got beyond "if," "then," and the occasional "but," plus I skipped that class all the time and ultimately dropped out because I preferred to smoke weed and play hackisack before lunch. (Oh, who am I kidding, I skipped all my classes, no matter the time of day. In my defense, it was some pretty good weed, and I was a ****in hackisack god. Fortunately, I squeaked by in a few classes because, believe it or not, I'm quite charming in person, and my teachers have mostly seemed to like me. I've been told I have charisma, whatever that means. If it was a girl who said it, it would probably mean "I'm ugly," but fortunately, it wasn't.) Aaaaaanyway, I thought logic was logic, and as you stated before, is consistent. What are these choices of which you speak, and how is that choice arbitrary? Do you care to provide any simple examples of what you just said?

What is arbitrary about logic is what sorts of statements we allow into the logic. Once we have defined the allowable statements, everything else is exact. So when applying logic to the real world, we need only make sure that we restrict ourselves to the allowable statements in the logic.

For example, in classical, first-order logic, the only allowable statement has the property that it is either true or false. Once you have that set up, the rest of the rules necessarily come about due to consistency: since the only allowable statements are true or false, a set of logic rules that leads to contradictory results is invalid.

In practice, this is how logical fallacies are discovered: we find a counter-example to the argument.

Finally, let me state that logic is just a way of thinking about the world. With logic, we take a series of propositions, and determine what can be drawn from those propositions. For example, if I take the propositions:
All boys have brown hair.
Bob is a boy.

...then I can infer that Bob has brown hair. Pure logic can never actually say whether the propositions or the conclusion(s) of a logical argument are true. But what it can do is link different propositions and conclusions together. In practice, we have to go out and look at the world to see whether or not our propositions or conclusions are true. For example, in the above case, if I look at Bob and find that he doesn't have brown hair, I now know that one of the two propositions must be wrong (either Bob is not a boy, or at least some boys don't have brown hair). The only uncertainty here is in my observation of Bob's hair color: I am equally as sure that one of the two propositions is wrong as I am sure that Bob doesn't have brown hair. There is no uncertainty in the logical deduction.

 

[Sage Lee]

Any time we apply mathematics to reality, we have to consider that we don't actually know for certain whether or not the mathematics applies.

In this situation, for instance, there's no such thing as a triangle in reality. You can draw something that looks like a triangle on a piece of paper with a pencil, for instance, but what it really is is a bunch of graphite and rubber atoms spread across the surface of the paper. It simply isn't possible to make atoms form a line segment, because the atoms are of finite size.

Because of this, it is very possible to draw something that looks like a triangle on paper, but doesn't actually have all of its properties.

Okay, makes sense

What is arbitrary about logic is what sorts of statements we allow into the logic. Once we have defined the allowable statements, everything else is exact. So when applying logic to the real world, we need only make sure that we restrict ourselves to the allowable statements in the logic.

For example, in classical, first-order logic, the only allowable statement has the property that it is either true or false. Once you have that set up, the rest of the rules necessarily come about due to consistency: since the only allowable statements are true or false, a set of logic rules that leads to contradictory results is invalid.

In practice, this is how logical fallacies are discovered: we find a counter-example to the argument.

Finally, let me state that logic is just a way of thinking about the world. With logic, we take a series of propositions, and determine what can be drawn from those propositions. For example, if I take the propositions:
All boys have brown hair.
Bob is a boy.

...then I can infer that Bob has brown hair. Pure logic can never actually say whether the propositions or the conclusion(s) of a logical argument are true. But what it can do is link different propositions and conclusions together. In practice, we have to go out and look at the world to see whether or not our propositions or conclusions are true. For example, in the above case, if I look at Bob and find that he doesn't have brown hair, I now know that one of the two propositions must be wrong (either Bob is not a boy, or at least some boys don't have brown hair). The only uncertainty here is in my observation of Bob's hair color: I am equally as sure that one of the two propositions is wrong as I am sure that Bob doesn't have brown hair. There is no uncertainty in the logical deduction.

Thanks, good explanation.

 

[Sage Lee]

I just stumbled across this thread:

https://www.physicsforums.com/showthread.php?t=59347&page=2

Where in post 19 someone talks about what I was trying to talk about but in a much more intelligent fashion. But I believe he points out that "it's only a consistent way of talking about reality because it misrepresents it" or something to that effect. Which still makes this all kind of pointless. Plus, that was 5 years ago, so maybe the works he's referencing have already been laughed off the table.

 

[Chalnoth]

I just stumbled across this thread:

https://www.physicsforums.com/showthread.php?t=59347&page=2

Where in post 19 someone talks about what I was trying to talk about but in a much more intelligent fashion. But I believe he points out that "it's only a consistent way of talking about reality because it misrepresents it" or something to that effect. Which still makes this all kind of pointless. Plus, that was 5 years ago, so maybe the works he's referencing have already been laughed off the table.

Well, while strictly correct in terms of mathematical/logical proof, what he wrote is very misleading. While we can never prove whether idealism or materialism is correct, we can obtain evidence that favors one or the other possibility. Materialism states that there exists a self-consistent reality external to ourselves which we perceive, however imperfectly. Such a reality, because it must be self-consistent, will contain patterns that allow us to make use of inference. Every time such inference is successful, we gain confidence that materialism is accurate. The success of modern science, then, provides a vast array of evidence in favor of materialism.

Idealism, on the other hand, which asserts that there is no way of knowing whether or not our putative observations are imaginary, possesses no such constraints. Imaginary worlds are not limited in any sense of the word, so that if we think we see some patterns, and make some predictions based upon those patterns, we may expect that sometimes those predictions will succeed, but usually they will fail, and if we wait long enough, those predictions will always fail, if idealism is accurate.

So when we have a scientific theory, such as Newtonian mechanics, that has stood the test of time, continually and repeatedly providing accurate answers to the same sorts of experiments, we have extreme confidence that idealism cannot be true.

We can never prove it, of course. This is the basic problem of inference. But the more our inference works, the more confident we are that it's a good way of doing things.

 

[GODISMYSHADOW]

Well, consider this by way of analogy.

The set of all even numbers is infinite. I can go on counting even numbers for ever and ever and never reach an end.

But clearly the set of all even numbers does not include all possible numbers. It doesn't include, for instance, the number pi.

So even if the universe is infinite (we don't know whether or not it is), then that doesn't necessarily mean that all possibilities are realized.

However, there may be other reasons to believe that all possibilities are realized, mainly stemming from quantum mechanics, where we find, for instance, that if there is the possibility of matter inhabiting a region of space, then particles of that sort of matter will necessarily pop in and out of the vacuum. Another way of saying this is that in quantum mechanics, there mere possibility of existence forces existence. So it is not unreasonable to suspect that perhaps all possibilities must actually be realized.

This doesn't mean that anything and everything we can imagine occurs, of course. We can imagine quite a lot of impossible things, as you mention above. But we can also imagine a great many things that are not obviously impossible, and yet may turn out to be upon deeper inspection.

Would the Universe be "the set of all things right now at this moment"? That can't be right, because Einstein showed there is no "absolute time" and hence no "absolute now". Could that mean there's really no Universe?

 

[marcus]

Would the Universe be "the set of all things right now at this moment"? That can't be right, because Einstein showed there is no "absolute time" and hence no "absolute now". Could that mean there's really no Universe?

"absolute" just means something that all observers agree on---it does not depend on the observer and his motion relative to other observers.

Just because you can have disagreement between observers (i.e no absolute time) doesn't mean the U doesn't exist.

However the phrase "right now at this moment" (that you used) does depend on what observers you are talking about----it takes some discussion.

The universe can exist just fine and yet different sets of observers can have different ideas about how to slice it into Present Moments.
==============================

I'll throw in some extra detail just in case anyone is curious to follow this further.

In cosmology we have a special set of observers!
A preferred perspective on the universe. So a preferred idea of simultaneity, and a time sometimes called "universe time" or "Friedmann model" time.

This set consists of all observers who are at rest relative to the ancient light.
The glow of ancient matter, from when the universe was just uniformly filled with hot gas. This glow is now the microwave background or "CMB".
An observer is at rest relative CMB if he perceives no big doppler hotspot ahead of him or coldspot behind. If he measures the temp approximately uniform in all directions.

We could have a network of observers all over the universe, all at rest relative CMB, and they could all synchronize their clocks! They could all agree on a slicing of events into synchronous slices. And they could all agree on the age of the universe.

Observers moving relative CMB would not agree, unless they compensated for their motion and took the viewpoint of a stationary observer.

And in fact that is what we do. We know the Earth's speed and direction relative CMB and we CORRECT observational data for that. We adjust so we can have data that is from the standpoint of a stationary observer. It is a very tiny correction because we are almost stationary, so in most situations you can neglect it.

But in a certain sense there is, in cosmology, a practical idea of an "absolute" time, or at least pragmatically preferred time, that the standard Friedmann equation model runs on, and corresponds to stationary observers time.

General Relativity allows this. The point is we have a kind of landmark. The glow from the ancient matter. Matter is what makes the difference.

 

[A. Neumaier]

In a real-world scenario, this would be like there being an observable universe somewhere far away that is absolutely identical to our own. If the universe is infinite, in fact, we know this must be the case, because due to quantum mechanics there are only a finite number of possible configurations. So if it is infinite in space, then the real universe would actually behave very much like the repeating number line, except that the repetition would be more chaotic than orderly.

From quantum mechanics we find that the total number of possible configurations of a given region of the universe is finite. It's a very large number, but a finite one nonetheless.

How did you arrive at this view?

How do you define a ''possible configurations of a given region of the universe'' in such a way that you can count their number?

 

[Chalnoth]

How did you arrive at this view?

Well, I thought I explained it sufficiently. Infinite space + finite configurations = repeating universe.

How do you define a ''possible configurations of a given region of the universe'' in such a way that you can count their number?

Well, there are a few ways to go about it. From one direction, we can approach the issue from the side of entropy, as entropy is proportional to the logarithm of the number of states that can replicate the macroscopic properties of the system (though this has the problem that we don't know how to calculate the entropy for every macroscopic configuration). From the other direction, we can approach the issue from quantum mechanics and just count the number of states that are available. This has the problem that we don't know the behavior at very high energies.

But in any event, the result, if we knew how to calculate it, would have to be finite in any case, because the entropy is finite and an infinite result for the quantum mechanical calculations would lead to nonsense in calculating simple things like reaction cross sections.

 

[A. Neumaier]

Well, I thought I explained it sufficiently. Infinite space + finite configurations = repeating universe.

I meant, why do you think that there are only finitely many configurations in an infinite universe?

From one direction, we can approach the issue from the side of entropy, as entropy is proportional to the logarithm of the number of states

the entropy is finite and an infinite result for the quantum mechanical calculations would lead to nonsense in calculating simple things like reaction cross sections.[/QUOTE]

A finite entropy density in an infinite universe may well lead to an infinite total entropy.

 

[Chalnoth]

A finite entropy density in an infinite universe may well lead to an infinite total entropy.

Yes, but we're not talking about total entropy, but rather the entropy of an observable region. And as long as the entropy density is finite, the entropy of an observable region of any given size will also be finite.

 

[A. Neumaier]

Yes, but we're not talking about total entropy, but rather the entropy of an observable region. And as long as the entropy density is finite, the entropy of an observable region of any given size will also be finite.

But the states in different observable regions may be different! Entropy doesn't tell you anything about that. (Otherwise, bu reducing the observable regions sufficiently, you could make the total number of distinct states as small as you like.

Moreover, there are vastly more states than the energy eigenstates counted by the entropy. Most observable systems are not in an energy eigenstate but in a complex superposition of these - and there are infinitely many possibilities for these, already for a single qubit.

 

[GODISMYSHADOW]

"absolute" just means something that all observers agree on---it does not depend on the observer and his motion relative to other observers.

Just because you can have disagreement between observers (i.e no absolute time) doesn't mean the U doesn't exist.

However the phrase "right now at this moment" (that you used) does depend on what observers you are talking about----it takes some discussion.

The universe can exist just fine and yet different sets of observers can have different ideas about how to slice it into Present Moments.
==============================

I'll throw in some extra detail just in case anyone is curious to follow this further.

In cosmology we have a special set of observers!
A preferred perspective on the universe. So a preferred idea of simultaneity, and a time sometimes called "universe time" or "Friedmann model" time.

This set consists of all observers who are at rest relative to the ancient light.
The glow of ancient matter, from when the universe was just uniformly filled with hot gas. This glow is now the microwave background or "CMB".
An observer is at rest relative CMB if he perceives no big doppler hotspot ahead of him or coldspot behind. If he measures the temp approximately uniform in all directions.

We could have a network of observers all over the universe, all at rest relative CMB, and they could all synchronize their clocks! They could all agree on a slicing of events into synchronous slices. And they could all agree on the age of the universe.

Observers moving relative CMB would not agree, unless they compensated for their motion and took the viewpoint of a stationary observer.

And in fact that is what we do. We know the Earth's speed and direction relative CMB and we CORRECT observational data for that. We adjust so we can have data that is from the standpoint of a stationary observer. It is a very tiny correction because we are almost stationary, so in most situations you can neglect it.

But in a certain sense there is, in cosmology, a practical idea of an "absolute" time, or at least pragmatically preferred time, that the standard Friedmann equation model runs on, and corresponds to stationary observers time.

General Relativity allows this. The point is we have a kind of landmark. The glow from the ancient matter. Matter is what makes the difference.

You're saying this "CMB" is used as a reference frame in your cosmology.
I'm going to have to study this stuff to gain a better understanding.

An event in the forbidden zone has no causal effect on my here-now because it's
outside the light cone. (That's absolute elsewhere on the Minkowski diagram.)
It's important to consider for astronomical distances. However, an event in the
forbidden zone may have a causal effect on some event happening in my future.
That being the case, if the universe is a set of events in the forbidden zone, then
the universe can't be more real than events in my future. That invites the question,
"Does the future exist?" Some say we can change our destiny if we try. Others say
the future is already there, it's irrevocable and cannot be changed. I wonder.

 

[A. Neumaier]

"Does the future exist?" Some say we can change our destiny if we try. Others say
the future is already there, it's irrevocable and cannot be changed. I wonder.

This is undecidable.

Suppose you'd record every detail about the history of the universe, wait till it has died, and then replay it in a perfect simulation (where of course, everything is already there). The physical laws would be exactly the same - without the slightest detectable difference.

 

[TrickyDicky]

...there is, in cosmology, a practical idea of an "absolute" time, or at least pragmatically preferred time, that the standard Friedmann equation model runs on, and corresponds to stationary observers time.
General Relativity allows this. The point is we have a kind of landmark. The glow from the ancient matter. Matter is what makes the difference.

Nicely put Marcus. I'd like to add to your insightful phrase in my bold that maybe sometimes we get hung upon abstractions about Spacetime, but it's good to remember ourselves once in a while that spacetime is just a geometrical abstraction to describe the relations within matter in its broad meaning of mass-energy continuum. In this sense matter is all there is and surely what makes the difference.

About the stationary observers, they illustrate the way the GR equations were designed in a general covariant way to have 6 independent differential equations with 6 unknown quantities and another 4 unknown quantities that are arbitrarily fixed with the choice of coordinates.
This condition allows us to stablish the rest frame or stationary observers as we set the coordinate space and the coordinate time for a particular metric, and therefore we can determine a rest state wrt these coordinates so in this sense the fundamental observers appear not only in the "Friedmann model" but in any metric we might build from the GR equations.

In our cosmological model this rest frame is embodied by the CMB like you say, we measure our motion with respect to this radiation that fills the vacuum thru the universe.

This is for a very practical reason, the CMB are photons and we are able to detect them, quite easily (from 1965 at least), we could say the CMB is the "visible" part of the energy density of the vacuum, which is indirectly observe or "felt" as dark energy (and also as dark matter according to some models with inhomogeneities such as those of T. Buchert et al., but these models are not mainstream).

 

[Chalnoth]

But the states in different observable regions may be different!

Yes, very true. So to do this properly, you'd have to integrate over all macrostates. That result, also, will have to be finite.

Moreover, there are vastly more states than the energy eigenstates counted by the entropy. Most observable systems are not in an energy eigenstate but in a complex superposition of these - and there are infinitely many possibilities for these, already for a single qubit.

The specific superposition of states is just a representational issue and thus cannot be a physical effect. That is to say, a particle that is in an eigenstate of energy is in a superposition of states in position. So you can recast any particle that we "see" as being in a superposition of states as being in a particular eigenstate by constructing your operator appropriately.

 

[A. Neumaier]

Yes, very true. So to do this properly, you'd have to integrate over all macrostates. That result, also, will have to be finite.

Nothing in quantum mechanics allows you to deduce this!

The specific superposition of states is just a representational issue and thus cannot be a physical effect. That is to say, a particle that is in an eigenstate of energy is in a superposition of states in position. So you can recast any particle that we "see" as being in a superposition of states as being in a particular eigenstate by constructing your operator appropriately.

But entropy only counts the eigenstates of the energy. On the other hand, most states in nature are not eigenstates (only stationary states are). Thus the vast majority of observable states is not counted by entropy.

 

[Chalnoth]

Nothing in quantum mechanics allows you to deduce this!

This stems from the exact same arguments as in quantum field theory: there has to be some high-energy cutoff.

But entropy only counts the eigenstates of the energy. On the other hand, most states in nature are not eigenstates (only stationary states are). Thus the vast majority of observable states is not counted by entropy.

Now you're mixing different descriptions of the same system. But it isn't true in any event. The computation of entropy has to count the full set of microstates, which for real particles also includes things like spin and angular momentum, as well as energy.

 

[Deuterium2H]

Coming full circle, and getting back to the original question/post...I think that my arguments using mathematically-based Set Theory, and Chalnoth's physics-based arguments (Thermodynamics, Statistical and Quantum Mechanics) have both converged on an answer that is rather non-intuitive. Certainly, it goes against popular "opinion". But if mathematics can teach us anything, it is that transfinite Set Theory is itself counter intuitive. This just so happens to be very much the case, as well, with Quantum Theory.

The answer to the the original post is quite simply this...

Given an infinite Universe, it is does NOT necessarily follow that "everything exists somewhere". Or, in other words, as previously argued...the Universe being infinite is a necessary condition, but not a sufficient condition to ensure that any/every event that has a finite probability must occur/exist somewhere in the Universe.

 

[A. Neumaier]

This stems from the exact same arguments as in quantum field theory: there has to be some high-energy cutoff.

Can you show why it should follow from that?

Now you're mixing different descriptions of the same system. But it isn't true in any event. The computation of entropy has to count the full set of microstates, which for real particles also includes things like spin and angular momentum, as well as energy.

If you look at the books of statistical mechanics, you'll find that microstates means only ''energy eigenstate'', and not ''arbitrary state''.

 

[Chalnoth]

Can you show why it should follow from that?

If the integration over macrostates is limited at some high energy, and every component of that integration is finite (that is, if the function is well-defined everywhere), then it will have to be finite, because it will be a representation of a sum over a finite (but large) number of states.

If you look at the books of statistical mechanics, you'll find that microstates means only ''energy eigenstate'', and not ''arbitrary state''.

I don't think this is true at all. The basis you do your sums in is completely irrelevant. It has to be, by nature of the underlying mathematics. The only reason why the sums are done in the energy basis is because:
1. Most introductory statistical mechanics books neglect complications like spin, angular momentum, and any other potential quantum numbers that are different from energy.
2. It is much easier to do the sums in terms of energy because the total energy of the system is one of the macroscopic variables we use.

In principle you could always change to some other basis, and if it's done right you have to come up with the exact same answer, but it's going to be much more difficult to connect the other basis to the macroscopic variables.

That said, this is an off-topic argument, because it simply has no application to my original statement, which had nothing whatsoever to do with entropy. Remember, I was making two separate points when talking about the finite number of potential states. The entropy argument was one argument, and is a separate one from the purely quantum-mechanical one.

The purely quantum-mechanical argument is that as long as you cut off your states at some high energy, there are a finite (though large) number of states. You came back and stated that you can also have superpositions of those states, and since there can be an infinite number of superpositions, this finite number of quantum states leads to an infinite number of possibilities.

Not so, I said, because the superpositions are merely a representational issue: any superposition of states can be represented as an eigenstate of the right operator. You'll still always get the exact same number of states, no matter the representation you use, as long as you do the counting correctly. This second argument for the finite number of states has nothing to do with statistical mechanics.

 

[A. Neumaier]

If the integration over macrostates is limited at some high energy, and every component of that integration is finite (that is, if the function is well-defined everywhere), then it will have to be finite, because it will be a representation of a sum over a finite (but large) number of states.

But this can be argued only locally. The energy cutoff of QFT is something at the level of individual scattering events, while the integration over macrostates in statistical mechanics never had such a cutoff.

I don't think this is true at all. The basis you do your sums in is completely irrelevant. It has to be, by nature of the underlying mathematics. The only reason why the sums are done in the energy basis is because:

No. The only reason why the sums are done in the energy basis is because the canonical ensemble involves the Hamiltonian, and the trace defining the entropy reduces to a sum _only_ in a representation where the basis states are energy eigenstates.

The purely quantum-mechanical argument is that as long as you cut off your states at some high energy, there are a finite (though large) number of states.

And I pointed out that both your hypothesis and your conclusion are flawed.

 

[Chalnoth]

But this can be argued only locally. The energy cutoff of QFT is something at the level of individual scattering events, while the integration over macrostates in statistical mechanics never had such a cutoff.

Typically you don't do any integration over macrostates in statistical mechanics. The integrations are over microstates. And you don't need any cutoff there because we are generally considering systems that are at such low temperatures that any cutoff that would come in from high-energy physics is irrelevant.

But when considering all possible states of the system, you have to integrate the number of states over the ensemble of all possible macrostates. As long as the number of states for any given macrostate is finite, and as long as you have to cut off your integral at some energy (so that the integral doesn't go to infinite), the result also has to be finite.

No. The only reason why the sums are done in the energy basis is because the canonical ensemble involves the Hamiltonian, and the trace defining the entropy reduces to a sum _only_ in a representation where the basis states are energy eigenstates.

And the reason why the canonical ensemble includes the Hamiltonian is because energy is one of the macroscopic variables. It is the only operator used because in the classical treatment, energy is the only thing that is allowed to be mixed (the particle number and volume tend to be fixed). When considering more complicated systems, such as a quantum system including spin or one where the particle number is allowed to vary, you have to make the ensemble a bit more complicated, so that it incorporates these added degrees of freedom.

It doesn't really matter, though. You can still transform to another basis if you like. The results will necessarily come out the same. It's just that the math will be horribly difficult, and thus it's much easier to just remain in the eigenbasis of your ensemble.

And I pointed out that both your hypothesis and your conclusion are flawed.

No, because you changed topics and started talking about statistical mechanics in an argument that had nothing to do with statistical mechanics.

 

[A. Neumaier]

No, because you changed topics and started talking about statistical mechanics in an argument that had nothing to do with statistical mechanics.

As if entropy and counting quantum states could be done without statistical mechanics.

 

[Chalnoth]

As if entropy and counting quantum states could be done without statistical mechanics.

Huh? Counting states is a component of statistical mechanics, but hardly requires it. Entropy doesn't even need to come into the argument when all you're interested in is the total number of possible states.

 

[Terraqueous]

About how long did it take for the quark-gluon plasma to cool?

I'm not completely sure on this, but I think the answer is 10^-6 seconds.

 

[GODISMYSHADOW]

This is undecidable.

Suppose you'd record every detail about the history of the universe, wait till it has died, and then replay it in a perfect simulation (where of course, everything is already there). The physical laws would be exactly the same - without the slightest detectable difference.

Are you suggesting the universe is a simulation?

 

[A. Neumaier]

Are you suggesting the universe is a simulation?

No, only that we couldn't distinguish it experimentally from a simulation. The physical laws would be exactly the same if the simulation was perfect.

 

[Tanelorn]

"No, only that we couldn't distinguish it experimentally from a simulation. The physical laws would be exactly the same if the simulation was perfect."


I am not sure I understand what you are actually saying there but I can say that there is a great difference between a mathematical simulation on a computer with a cpu executing single arithmetic instructions one bit at a time and the space time reality we are part of. In a similar way it is highly unlikely that life like intelligence can ever be created on such a simple calculating device.

 

[A. Neumaier]

"No, only that we couldn't distinguish it experimentally from a simulation. The physical laws would be exactly the same if the simulation was perfect."


I am not sure I understand what you are actually saying there but I can say that there is a great difference between a mathematical simulation on a computer with a cpu executing single arithmetic instructions one bit at a time and the space time reality we are part of. In a similar way it is highly unlikely that life like intelligence can ever be created on such a simple calculating device.

Of course. If our universe were a simulation, it would have been simulated on one of God's hyper-computers with a very different physics and technology.

The point is, we couldn't see the difference in the results.

 

[Tanelorn]

Or the Universe and God could be one and the same thing. No simulation required :)

 

[GODISMYSHADOW]

Of course. If our universe were a simulation, it would have been simulated on one of God's hyper-computers with a very different physics and technology.

The point is, we couldn't see the difference in the results.

So many different views! To me, the universe is a probability with no
provable objective reality.

 

[Tanelorn]

Or the Universe and God could be one and the same thing. No simulation required :)

After hearing Anthony Hopkins discuss his support yesterday of the Philosopher Spinoza's views I decided to dig a little deeper and was pleasantly surprised that I share many of the sentiments:


Albert Einstein named Spinoza as the philosopher who exerted the most influence on his world view (Weltanschauung). Spinoza equated God (infinite substance) with Nature, consistent with Einstein's belief in an impersonal deity. In 1929, Einstein was asked in a telegram by Rabbi Herbert S. Goldstein whether he believed in God. Einstein responded by telegram: "I believe in Spinoza's God who reveals himself in the orderly harmony of what exists, not in a God who concerns himself with the fates and actions of human beings." Spinoza's pantheism has also influenced environmental theory; Arne Næss, the father of the deep ecology movement, acknowledged Spinoza as an important inspiration.


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


I apologise if this is overly philosophical, I will not add to this, I just thought it was an interesting comment.

 

[Chalnoth]

Yeah, I personally never liked that idea as it always seemed to me that "God" carried with it far too much anthropomorphic meaning to be anything but misunderstood when used in that way. It sounds like an attempt to re-purpose the religious word to describe some feeling of awe or wonder regarding the universe itself. But I just don't see the purpose in doing that. Can't we describe the majesty of the universe without resorting to anthropomorphic words? And there remains, to me, a significant downside in that the religious merely use it as an excuse to trumpet their own views (the religious absolutely love to imagine that science is on their side, and famous scientists talking about "God" are exceptionally tantalizing).