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

[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).

 

[Tanelorn]

Chalnoth, I sympathise with your views also. In fact I find I can move between Atheism, Agnosticism and Pantheism, sometimes all on the same day. Perhaps in his statement Einstein was helping by leading people from the old superstition anthropomorphic based religions into a higher state of enlightment, taking baby steps so to speak. Hopefully we will avoid the fate that Sagan was so concerned about. The main reason I have for sometimes believing in something greater is that it sometimes appears to me that there was a very powerful and intentional force behind the creation of the universe. It can't be proven, but the universe seems so finely tuned, too much so for random chance. The whole thing seems so unlikely, and instead we could have had a universe consisting of nothing more than an infinite amount of green jelly!

In the Anthony Hopkins interview, a fellow Welshman, I particularly agreed with his views regarding people "who know the truth". Such certainties gave rise to people like Hitler with plans for everyone. I have come to similar conclusions myself.

 

[Deuterium2H]

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.

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.


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.")

Hi Sage,
Sorry for resurrecting this older thread, but I happened to be re-reading through it for another reason, and had previously missed a statement you made, in error, that may cause all sorts of confusion if left uncorrected. The Cardinality of the Empty Set (Null Set) is not one, it is zero. The Set that contains the Empty Set is equal in Cardinality to one. In fact, in axiomatic Set Theory (e.g. ZFC), the existence of the Empty Set is defined as fundamental Axiom. It is upon this, and the following Pair Set and Sum Set axioms that larger Sets are created...thusly:

{ }= ø = 0
{{ }} = {ø} = 1
{{{ }}} = {ø,{ø}} = {0,1} = 2
{{{{ }}}} = {ø,{ø},{ø,{ø}}} = {0,1,2} = 3
etc., etc.

 

[nakian]

I read some answers that tended to argue that the possibility that everything could exist was unlikely. Other comments gave the impression that having a twin in another world sounded like sci-fi... Maybe you should spend some times reading what Max Tengmark has to say about the Multiverse http://space.mit.edu/home/tegmark/PDF/multiverse_sciam.pdf. Also find out more here http://en.wikipedia.org/wiki/Multiverse#Level_I:_Beyond_our_cosmological_horizon.

The argument Tengmark makes is that worlds similar to ours are very likely, that is the likelihood that you have a twin somewhere in another world is high. Those un-observable universes , those of level-I, that is worlds beyond our cosmological horizon, will probably be of an infinite number. They will all have the same physical laws and constants as ours. Everything that is possible in our world will be possible in those worlds. In that sense, everyhing that could happen here, even if it will never happen here or has never happened here, would probably have happened or will probably happen somewhere in a Level-I un-observable world. In conclusion, it is highly probable that you have a twin somewhere, dating J-Lo's twin in that world...

 

[nakian]

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. interesting comment.

By impersonal do we mean "unconscious"? Because, Spinoza God also possesses the Attribute of being infinitely conscious. I am not to sure what being infinitely conscious means, but I pretty sure it's not the same thing as "unconscious". Am I confusing things here?

 

[Tanelorn]

nakian, welcome to PF!

Also I thank you for the multiverse links and question about Spinoza.

These subjects are very interesting to me, however they are also highly speculative so we may need to discuss them elsewhere. This Cosmology forum is meant for questions on the hard science of the standard model, but the thread seems to have survived thus far.

This paper on Spinoza is interesting. On pages 23 and 24 there is discussion on Spinoza's view of conciousness:
http://philosophy.fas.nyu.edu/docs/IO/2575/garrett.pdf

I presume you also enjoy the works of Nakian?

 

[Tanelorn]

By impersonal do we mean "unconscious"? Because, Spinoza God also possesses the Attribute of being infinitely conscious. I am not to sure what being infinitely conscious means, but I pretty sure it's not the same thing as "unconscious". Am I confusing things here?

By "impersonal deity" Einstein could be meaning one or more of the following.

1. not personal; without reference or connection to a particular person: an impersonal remark.
2. having no personality; devoid of human character or traits: an impersonal deity.
3. lacking human emotion or warmth: an impersonal manner.

I agree though that Spinoza's God is infinitely conscious, whereas Einstein seems to be saying that his God is devoid of human character, traits and personality.

More recently some may have also relegated God further, to a God of nature, an unconscious force of creation. Lovelock's Gaia principle may also be related to this view of God. ie. A Gaiaverse.