Please explain it to me. If there are no measure zero events, then NO distribution of states to universes is possible. Just like if you flip infinitely many coins. Whatever result comes up, that was a measure zero event. — fishfry

Do we have to go back to Kindergarten? If we do, I'm feeling like the much anticipated demolition of Cosmic Inflation has a somewhat uninspiring probability.

What strikes me as particularly strange, is that it seems you think it possible, in a Hubble Volume, to flip a coin infinitely many times, and presumably store the result? I am literally suppressing laughter.

I think maybe you should either accept the physics or try to understand it.

An ergodic random field whose harmonic oscillator coefficients span {1,2,3,4,5,6} with a Gaussian distribution centred on 3.5. The field is in a state of superposition and decoheres for infinite time. What is the probability that the decoherence branch with initial condition "4" will be encountered in the multiverse?

Can you provide a reference to the derivation?

I think maybe you should either accept the physics or try to understand it. — tom

If all you have is insults and you can't coherently explain your point, you've got nothing. You're the one claiming there are infinitely many universes and/or finite regions of an infinite space. My analysis applies.

You earlier claimed that it's been established in this thread that measure zero events can't occur. Two or three people including myself have asked you for the explanation. You have not provided one.

You can go look up the formal definition of ergodicity to learn that the smooth statistical behavior may fail to occur on a set of measure zero. This is a required characteristic of any probabilistic argument, since events on a set of measure zero have no effect on the overall probability.

I have not noticed such an explanation. But it's a long thread and I haven't read it all. Can you please point to one such explanation? — andrewk

I repeat. All possible initial states are instantiated over a Gaussian distribution due to the nature of the Inflaton.

There is no member of the initial superposition of all possible states of the Inflaton that is singled out for non-existence.

Physics is Unitary.

After finite time there are a countably infinite number of indistinguishable Hubble Volumes, which, as time progresses, may diverge.

There are measure zero sets in the Gaussian distribution (and in Gaussian Random fields) since their (finite state) distributions are continuous with respect to the Lebesgue measure, tho. :(

Can you provide a reference to the derivation? — fdrake

What? Are you joking? You need a derivation and a reference? You are giving me a headache.

Yes. If you're going to say a result is established in physics, and is obvious. It should come with either a reference to either the paper or popular science article that establishes it, or a description of the text which suggests it.*

It isn't at all obvious that @andrewk and @fishfry's comments regarding measure zero sets have been addressed, but I am taking it on faith that you know the derivation or where to find it. The alternative is you don't know the derivation or the behaviour, or a reference to either, and your incredulity at our questions is unfounded.

*if requested.

After finite time there are a countably infinite number of indistinguishable Hubble Volumes — tom

I don't understand how you can say this, yet claim you don't understand the idea of countably many coin flips. They're the same mathematical idea.

You have countably many regions, or universes, or coin flips. Each region or universe or flip is assigned one out of finitely many possible states. One out of a zillion in the case of physics, or one out of 2 in the case of coin flips, but the math is exactly the same either way.

You can't mock the idea of coin flips and then come back with the exact same idea in the guise of countably many universes. There's no mathematical difference between a 2-sided coin or a gazillion-sided coin. If the number of states is finite, then probability theory applies. In the large, it is "almost certain" that all states recur infinitely many times, but it is not absolutely certain. The case of coins or universes are exactly the same. It only depends on there being countably many coins or universes or regions, each taking up one out of at most finitely many states.

The coins are a simplification for the purpose of making the probability theory clear. But I can use a gazillion-sided die if you like, where the number of possible arrangements of particles is a gazillion. The math is absolutely the same.

Yes. If you're going to say a result is established in physics, and is obvious. It should come with either a reference to either the paper or popular science article that establishes it, or a description of the text which suggests it.* — fdrake

What? It's a toy example! If you need a reference, you need your head examining. Seriously, I am feeling vicariously embarrassed and ashamed, and somewhat defiled.

Ok then.

I don't understand how you can say this, yet claim you don't understand the idea of countably many coin flips. They're the same mathematical idea. — fishfry

Umm, did I ever claim incomprehension?

You have countably many regions, or universes, or coin flips. Each region or universe or flip is assigned one out of finitely many possible states. One out of a zillion in the case of physics, or one out of 2 in the case of coin flips, but the math is exactly the same either way. — fishfry

Umm, no. There are uncountably many Hubble Volumes instantiating countably many initial states, at least that is what the theory says. What is a "zillion"?

You can't mock the idea of coin flips and then come back with the exact same idea in the guise of countably many universes. There's no mathematical difference between a 2-sided coin or a gazillion-sided coin. If the number of states is finite, then probability theory applies. In the large, it is "almost certain" that all states recur infinitely many times, but it is not absolutely certain. The case of coins or universes are exactly the same. It only depends on there being countably many coins or universes or regions, each taking up one out of at most finitely many states. — fishfry

You make me laugh.

After finite time there are a countably infinite number of indistinguishable Hubble Volumes, which, as time progresses, may diverge. — tom

Umm, no. There are uncountably many Hubble Volumes instantiating countably many initial states — tom

[My bolding in both quotes]

In one post you said there are countably many Hubble volumes, and in another post you said there were uncountably many. Can you clarify this?

We would all benefit if you would please lay out a clear version of your argument in one place for reference. Assumptions, argument, conclusions. So we can know we're all talking about the same thing.

I perfectly well admit that I'm ignorant of physics. If someone can explain to me why probability theory doesn't apply to this situation I'd be happy to learn.

My understanding of your argument is that at the moment of the creation of the multiverse every possible state gets instantiated.

As I understand it, the argument for that conclusion is probabilistic. It depends on the behavior of statistical distributions at the limit. You used that phrase as one of your bullet items earlier. You are making a probabilistic argument to demonstrate that it's almost surely the case that every state gets instantiated. And you don't understand that your argument allows for the possibility of a large set of exceptions.

I'm asking for a clear explanation for why the well-known rules of probability don't actually apply to the creation of the multiverse, when these very rules are employed to develop the theory. I'm not saying there's no such explanation. I'm asking what the explanation is. Then I'll learn something.

I've heard that Tegmark himself admits he's no mathematician. I wouldn't be surprised if he's just ignored this point and nobody cares much. It's not that big a deal. As I said earlier, what if the multiverse got started and all but one state were instantiated. It would still be ergodic. Can you tell me the exact reason you claim that can't happen? If it happened, would the universe collapse or would we all have three heads? My intuition doesn't see why it couldn't miss a state yet still satisfy all the relevant statistical properties.

In one post you said there are countably many Hubble volumes, and in another post you said there were uncountably many. Can you clarify this? — fishfry

Could you just read the quotes again, slowly, and out loud?

Or, try swapping the parts in bold:

There are a countable infinity of INDISTINGUISHABLE Hubble Volumes, which diverge.

There are uncountably many Hubble Volumes, instantiating countably many initial states.

My understanding of your argument is that at the moment of the creation of the multiverse every possible state gets instantiated.

As I understand it, the argument for that conclusion is probabilistic. — fishfry

No, the physics is Unitary, all the way back to the Inflaton, and beyond.

There are a countable infinity of INDISTINGUISHABLE Hubble Volumes, which diverge. — tom

You might explain for us hoi polloi how indistinguishable things can be counted, because we would have thought that distinguishing something is a prerequisite for counting it.

For those interested, the argument that, as a generic consequence of inflationary cosmology, there almost certainly exist exact duplicates of Earth (among other interesting things) is given here: Many worlds in one, J. Garriga, A. Vilenkin, Phys.Rev. D64 (2001). (This is still within the parameters of "level-I multiverse.") — SophistiCat

Hi Sophisticat. I skimmed that article you linked and was interested to note that Vilenkin makes statements like:

"there are an infinite number of O-regions with identical histories up to the present"

where I think what he means is "there is almost surely an infinite number of .....". That is, I think he over-simplified his statement, presumably because he wanted to make it more accessible to the non-physicist reader, since it is a non-technical article.

I note that in your post you included the crucial qualifier "almost certainly", although it does not occur in the paper. Interestingly, Tegmark also omits the qualifier (bottom of first column on page 4 of this article you linked) but, like Vilenkin, gives no explanation for the omission, and his article is also more pop science than academic.

Do you have a view on why they omitted the 'almost certain' qualifier from their articles?

You might explain for us hoi polloi how indistinguishable things can be counted, because we would have thought that distinguishing something is a prerequisite for counting it. — Wayfarer

I didn't say you could count them. You can't count them.

where I think what he means is "there is almost surely an infinite number of .....". That is, I think he over-simplified his statement, presumably because he wanted to make it more accessible to the non-physicist reader, since it is a non-technical article. — andrewk

This is truly weird. You think you know better than Garriga and Vilenkin what they actually mean?

In Vilenkin's book, which he published 6yrs later, he repeats that paragraph verbatim. He also writes:

"Yes, dear reader, scores of your duplicates are now holding copies of this book. They live on planets exactly like our earth, with all its mountains, cities, trees and butterflies. The Earths revolve around perfect copies of our Sun, and each Sun belongs to a grand spiral galaxy - an exact copy of our Milky Way."

Also:

"It follows from the theory [of eternal inflation] that island universes are infinite and that the initial conditions at the big bang are set by random quantum processes during inflation. The existence of clones is thus an inevitable consequence of the theory."

You might explain for us hoi polloi how indistinguishable things can be counted, because we would have thought that distinguishing something is a prerequisite for counting it.
— Wayfarer
I didn't say you could count them. You can't count them. — tom

You're being illusive. Wayfarer has a point, and you know the next question.
Countable means you can assign a number to any of these volumes, and to do that they must be distinguished. If they can't be, they're not countable. So volumes like ours, we're (arbitrarily) 1 and the nearest copy is 2. It is distinguished by being nearest. Of course the copy of me over there considers his volume to be 1 and ours (of which he is unaware) to be 3 because there happens to be another one closer to him. We're both unaware of the actual locations of those copies, so identical state is maintained. It is an objective ordering. Anyway, they're identical in state, but still objectively distinguishable and thus objectively countable.
I think it is a mistake to say this then:

There are a countable infinity of INDISTINGUISHABLE Hubble Volumes, which diverge. — tom

They must be distinguishable but have at least identical state. If identical state, how can they diverge? You must consider the full set of worlds as the one state, else there is no 'current state' with which another volume can be identical. To do so presumes a QM interpretation like Copenhagen with real chance and action at a distance and a bunch of baggage that muddies the statement that the two volumes are actually identical.

Heck, with that definition, maybe every Hubble Volume is identical state, and yes, that would mean uncountable.
Sorry for this post. Just thinking out loud.

You're being illusive. Wayfarer has a point, and you know the next question.
Countable means you can assign a number to any of these volumes, and to do that they must be distinguished. If they can't be, they're not countable. — noAxioms

No it doesn't. You can't count your clones. Physics tells us that the cardinality of your clones is Aleph_0.

If you think it is possible to count your clones, I urge you to try.

If identical state, how can they diverge? You must consider the full set of worlds as the one state, else there is no 'current state' with which another volume can be identical. — noAxioms

I said the Hubble Volumes are INDISTINGUISHABLE not identical. I also mentioned somewhere that the Hubble Volumes would be INDISTINGUISHABLE after FINITE time.

The distinction is that the Volumes have the same history, not necessarily the same future.

The distinction is that the Volumes have the same history, not necessarily the same future. — tom

If they have the same history, and if determinism is the case, then wouldn't they also have the same future?

Edit: I'm sure I recall you claiming to be a determinist. If not, then I suppose this question can be ignored.

If they have the same history, and if determinism is the case, then wouldn't they also have the same future? — Michael

That's why I brought up QM interpretations.
Copenhagen is in theory nondeterministic, but involves faster than light action, and thus destroys the limits of the Hubble volume. It also leaves states in superposition if not measured from an arbitrary observation point. Pull back the camera so to speak and Copenhagen becomes as deterministic as any other interpretation.

No it doesn't. You can't count your clones. Physics tells us that the cardinality of your clones is Aleph_0.

If you think it is possible to count your clones, I urge you to try. — tom

I did in the post to which you replied. Perhaps you think that countable means you can know how many there are, but then the integers are not countable, so you're working from a different rule book.

I said the Hubble Volumes are INDISTINGUISHABLE not identical.

I think you need to expand on what you mean by these terms since we seem to be talking past each other.

If they have the same history, and if determinism is the case, then wouldn't they also have the same future? — Michael

If you perform a quantum measurement - e.g. a measurement of z-spin of a particle prepared in x-spin-up configuration, and choose your spouse based on the result, in half your futures you are married to Mary, in the other half it's Jane. Same past different futures.

Determinism is dead. Long live Unitarity!

If you perform a quantum measurement - e.g. a measurement of z-spin of a particle prepared in x-spin-up configuration, and choose your spouse based on the result, in half your futures you are married to Mary, in the other half it's Jane. Same past different futures.

Determinism is dead. Long live Unitarity! — tom

I thought you pushed the view that you're married to both of them, a deterministic view.
I just now see Michael's edit where he notes the same view shift.

I thought you pushed the view that you're married to both of them, a deterministic view.
I just now see Michael's edit where he notes the same view shift. — noAxioms

In half your futures you are married to Jane. In the other half, you are married to Mary. Are you suggesting I have ever promoted polygamy?

Of course I am a determinist, it's just that the picture of how determinism applies in Reality has to take into account quantum mechanics. This is different from classical determinism.

Let's be rigorous about our definition of the state of a type-1 universe, that with which we would need to have copied.

I think all you need is a specification of an event, frame, and state. The frame is there for future state only, not present state. So:

Earth CoG at stroke of midnight Y2K, in the arbitrary frame of Earth CoG at that moment, in the state that is the direct causal history of me making this post 17 years later. That defines a temporal line of that point in space that goes out forever. A different Hubble Volume is identical to that one if the wave function of any point along that line is the same in both volumes.
Does that work? It totally leaves out the definition what is the 'current state' of our volume, and I think it might even be independent of QM interpretation.

I have recently concluded that whether or not multiverses are real is not a very interesting question since currently we have no way of observing the truth of it.
However, I do find that considering reality in 10 dimensions, the 7th and beyond relating to multiverses, is a useful tool when determining the characteristics of the universe we can observer by comparing it to the arbitrary characteristics of other "possible" universes.

I skimmed that article you linked and was interested to note that Vilenkin makes statements like:

"there are an infinite number of O-regions with identical histories up to the present"

where I think what he means is "there is almost surely an infinite number of .....". That is, I think he over-simplified his statement, presumably because he wanted to make it more accessible to the non-physicist reader, since it is a non-technical article.

I note that in your post you included the crucial qualifier "almost certainly", although it does not occur in the paper. Interestingly, Tegmark also omits the qualifier (bottom of first column on page 4 of this article you linked) but, like Vilenkin, gives no explanation for the omission, and his article is also more pop science than academic.

Do you have a view on why they omitted the 'almost certain' qualifier from their articles? — andrewk

Well, I cannot answer for Vilenkin or Tegmark, but I think they were speaking informally.

How we interpret these results depends on how we think about probability. If we interpret probability as a quantitative measure of credence, or degree of belief, then there isn't really a difference between "almost surely" and "surely": in either case, the credence is exactly zero. This failure to make a distinction between possibility and impossibility may be a deficiency of the epistemic interpretation of probability (not to mention the problems of formal probabilistic modeling that have been raised here).

But if we further think about our concepts of probability and possibility, this might be argued to be a distinction without a difference. We can hardly tell the difference in credence between an event that has a probability of 10^-10 in a single trial and one with a probability 10^-100. We stop making a difference long before "almost surely".

There is still a possible/impossible distinction though. But is there, really? If "an event A is impossible" means for you that you should live your life as though A will never happen, then events with an extremely low probability are as good as impossible. You live your life assuming that the air will not suddenly evacuate the room through the window, leaving you choking on the floor, even though science says that such an event is possible (and even has a well-defined, finite probability!)

There is still a possible/impossible distinction though. But is there, really? If "an event A is impossible" means for you that you should live your life as though A will never happen, then events with an extremely low probability are as good as impossible. You live your life assuming that the air will not suddenly evacuate the room through the window, leaving you choking on the floor, even though science says that such an event is possible (and even has a well-defined, finite probability!) — SophistiCat

Right, but if we're thinking about the universe at large, then all these low probability events could be happening elsewhere, assuming a large enough universe. There would even be worlds where the low probability events are common, and assuming anything intelligent can manage to survive long enough there, it might come to a different conclusion about how low those probabilities are.

In an infinite universe, aren't we almost surely guaranteed a world where our doppelgangers walk through walls (the molecules align just right) after saying an incantation? Maybe doppleganger Jesus really did take a stroll on the water.

The case of a simple bound system, such as a hydrogen atom, is easier to analyze than a more general case: we can actually solve the quantum equations and enumerate every possible state. There is, however, a theorem for the general case in quantum mechanics, which puts a limit on the number of possible states, or degrees of freedom, given a volume and energy density within that volume.

The general point that I wanted to make is that if there are separate systems with a finite number of possible states between them, then for them to be found in the same state at some moment, they do not have to have identical histories up to that moment. Even in a purely deterministic universe, as these systems transition from one state to another, they may end up in the same state at some point simply by chance. What that chance is - high, low, "almost surely" - will depend on a more detailed analysis.