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

I've quoted from Vilenkin's book. Nerither he not Tegmark 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). — SophistiCat

It's nothing to do with probability, or our interpretation of probability. All initial conditions are realised by inflation, deterministically.

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". — SophistiCat

It's nothing to do with credences either.

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

Impossible events are those forbidden by the laws of physics.

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 — SophistiCat

And, this is also true for Hubble Volumes in the Level 2 Multiverse. Indistinguishable Hubble Volumes will have different histories due to different laws of physics operating.

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. — SophistiCat

Not sure which post brings on this reply. I brought up an insanely complex quantum equation in my prior post, but never suggested it was in need of being expressed or solved.

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.

We're talking a hubble-volume in this case, which has a finite but large degree of freedom. My wave function was based on that. Interestingly, I think it was a mistake to specify an inertial frame in my description. The full wave function of the one event is enough. If another event somewhere has the same wave function, it defines a clone Hubble sphere to ours.
It is also a definition free from tom's concern about the two universes staying identical. The definition is of an event which doesn't become something else.
My definition breaks down with Bill Clinton's oddly applicable statement: "It depends on what your definition of 'is' is". How can anybody assert that the state of some event outside our sphere 'is' in any particular state? Our definition sort of assumes a measurement taken from 'here', and by that definition, those distant events have no measurement and are in complete superposition. The nearest Earth clone is massively closer than the figure Tegmark quotes where the number of finite states is computed and divided by distance, something that seems invalid without a measurement being taken, from here no less.
The distance then becomes a function of the furthest historic matter that made a difference to our state now. That's further out that the Hubble-Volume, which is defined as the stuff that can make a future causal difference to here, not that which has made a past causal difference.

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.

Agree with this. Yes, I think I alluded to the opposite at first, but you're right. This was pointed out to me in a prior post.

I've quoted from Vilenkin's book. Nerither he not Tegmark were speaking informally. — tom

They do not say that they are speaking precisely and formally in their books. It is only you that says that. The evidence points to the opposite being the case. The absence of equations is a big clue.

In any case, the books are not holy scripture and we are not in the helpless position of those trying to interpret holy scripture and work out what the Author intended. Either mathematical analysis supports a conclusion that there does not exist a single level 1 spacetime lacking a duplicate Earth, rather than the set of such spacetimes merely having measure zero, or it doesn't. If it does, you should be able to point to a rigorous proof of the former. So far you have not done so.

I think I found the smoking multiverse.

I went to the Wiki article on the multiverse. I searched for "ergodic," and found this:

A prediction of chaotic inflation is the existence of an infinite ergodic universe, which, being infinite, must contain Hubble volumes realizing all initial conditions.

Now we see a couple of things. First, here is the popularized explanation of the ergodic business. But note the imprecision and inaccuracy:

... which, being infinite, must contain Hubble volumes realizing all initial conditions.

The language here clearly shows that the author thinks that the mere fact that there are infinitely many Hubble volumes implies that all initial states must be instantiated. But of course we have seen that this is not true.

I do realize that the fault here is not with Tegmark or with any other serious multiverse theorists, but rather with the Wiki author. But this is a good example of the type of loose thinking and careless use of infinity that seems to be a big part of this subject.

Now two paragraphs later, we find this beautiful resolution to our entire problem.

Given infinite space, there would, in fact, be an infinite number of Hubble volumes identical to ours in the universe. This follows directly from the cosmological principle, wherein it is assumed that our Hubble volume is not special or unique.

AHA!!!! Mere infinity is not sufficient, we all agree on that. And as I've pointed out, ergodicity is not sufficient, because a distribution could be ergodic yet still behave badly on a set of measure zero.

We need another assumption. the cosmological principle, which says in effect that there are no measure zero misbehaviors!

So ergodicity says that the large scale behavior is statistically well-behaved; and the cosmological principle says that there are no measure zero exceptions. Ergodicity allows that for all we know, our own earth is a statistical exception. The cosmological principle says that there are no statistical exceptions.

None of us know (though some claim to know) how the world got started, if was really a big bang multiverse or whether it's just turtles all the way down.

But as philosophers we can at least point out which arguments correctly follow from which assumptions.

As far as I can determine, the fact that there must be a duplicate earth does not follow from ergodicity; but rather from the additional assumption of the cosmological principle. It's the cosmological principle that rules out measure zero exceptions. And of course the cosmological principle is an assumption not directly supported by evidence. It's simple an assumption whose purpose is to make the duplicate earth theory true. Without that extra assumption you haven't got certainty that there's a duplicate earth.

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

This is something that troubles me deeply. You are claiming that physicists assume that Zermelo-Fraenkel set theory applies to the physical universe.

If that were true, wouldn't their be clever physics postdocs applying for grants to see if the axiom of choice or the continuum hypothesis or any of the large cardinal axioms are true?

If you claim that the collection of clones may be placed into bijection with the set of positive integers, then how large does physics say is the collection of the set of subsets of the clones? Is it Aleph-1, as the continuum hypothesis would imply? Or do you think it might be Aleph-2, as ‎Gödel suspected? Or perhaps far larger than that, as Paul Cohen believed?

It seems to me that these questions are utter nonsense. Nobody knows whether ZF applies to the real world. Perhaps category theory or homotopy type theory, two modern alternatives to set theory, better describe the mathematical foundations of the world.

You have a reference for the claim that physicists believe there are countably many clones? Or that Zermelo-Fraenkel set theory has relevance to physics? This I would really like to see.

Along the same lines, if you claim there are uncountably many indistinguishable Hubble volumes, what is the cardinality of this uncountable quantity? Is it one of the Alephs? Or perhaps the axiom of choice false and the uncountable cardinality is not an Aleph at all. Could that be the case?

I don't think physics has given any credence to these questions at all. In which case any claim about the cardinality of clones is nonsense.

We need another assumption. the cosmological principle, which says in effect that there are no measure zero misbehaviors! — fishfry

I think the cosmological principle allows such exceptions, but just says that the probability of us being that exception is sufficiently infinitesimal to preclude explanations that require us to be that exception.

As such, it is, as you say, an assumption, not some mathematical certainty that we're not unique.

I think the cosmological principle allows such exceptions, but just says that the probability of us being that exception is sufficiently infinitesimal to preclude explanations that require us to be that exception. — noAxioms

Well the probability is zero. But that is not the same as guaranteeing with absolute certainty that there is another earth. That's the claim on the table. The difference between absolutely certain or almost certain. If someone wants to claim that the multiverse theory says that it's almost certain [in the technical sense] that there's a duplicate earth, I'll accept that this conclusion follows from the premises. But @Tom is claiming absolute certainty, and I don't see it.

We need another assumption. the cosmological principle, which says in effect that there are no measure zero misbehaviors! — fishfry

Well actually the author has misused the cosmological principle, which implies nothing of the sort. The cosmological principle states that each constant-time hypersurface of the universe ('this spacetime') is homogeneous and isotropic at the large scale. When formalised (which is quite tricky to do - see this discussion), this is a statement about observed average quantities as the size of the hypersurface subsets we average over approaches infinity.

So the cosmological principle says absolutely nothing about microphenomena such as whether a particular teensy-weensy arrangement of molecules like the Earth recurs - even though the author of that article appears to think it does. I am confident that neither ergodicity nor the cosmological principle, either alone or together, can imply the conclusion that there is certainly a duplicate Earth in this spacetime.

So the cosmological principle says absolutely nothing about microphenomena such as whether a particular teensy-weensy arrangement of molecules like the Earth recurs. — andrewk

I suspected that, but didn't want to complicate my point.The Wiki article on the cosmo principle does note that the sun is different from the earth, so that the cosmo principle doesn't apply at such small scales. But if there are two Hubble volumes that are identical, that must (might?) mean that those volumes are particle-by-particle identical, which would imply a duplicate earth. Is that true? If there are two identical Hubble volumes, what does identical mean? Are they quark-by-quark the same? Ih which case they have duplicate earths.

I must say that I don't really believe there are duplicate earths, duplicate people, or for that matter an actual infinity of anything. I'm with Lee Smolin, who coined the phrase, "the trouble with physics." He was talking about string theory, but multiverse theory strikes me as suffering from the same type of problem. We are at the limit of our ability to do experiments, so the theorists are running amok and no longer doing science.

https://www.amazon.com/Trouble-Physics-String-Theory-Science/dp/061891868X

The cosmological principle states that each constant-time hypersurface of the universe ('this spacetime') is homogeneous and isotropic at the large scale. — andrewk

Aren't you neglecting that the matter density must be uniform? You have to count the contents too. Spacetime won't be flat unless the matter is presumed to be evenly spread.

.The Wiki article on the cosmo principle does note that the sun is different from the earth, so that the cosmo principle doesn't apply at such small scales. — fishfry

And remember that Linde's eternal inflation would presume that each bubble universe would start off at a planckscale energy density and so the initial state would be a relativistic gas, a quark-gluon hot soup. So the material content would be at thermal equilibrium. The only fluctuations - the seed forming inhomogeneities that result in the later gravitational/material structure - would be thermal quantum ones.

So both protons and electrons, stars and galaxies, are local inhomogeneities that pop out way after any such structure has been washed clean by an initial thermal equilbration.

Of course that then is a constraint on the odds of the history of a universe actually repeating "particle for particle". However no one wants to talk about the real combinatorial issues here. :)

but multiverse theory strikes me as suffering from the same type of problem — fishfry

It's possible to dislike multiverse hypotheses but not blame it on physics, because it's all unfalsifiable and hence doesn't count as science. I regard it as metaphysics.

Smolin's complaint is that the same applies to string theory. If he's correct (I don't know enough about string theory to comment) then string theory also is not science and so should not be getting large parts of physics funding. Also, it should be called 'String Hypothesis', as a requirement of any 'theory' is that it be falsifiable.

Aren't you neglecting that the matter density must be uniform?

The homogeneity part of the cosmological principle requires that mass-energy be uniformly distributed 'at the large scale'.

It still has to start off homogenous and thermalised at the small scale of the initial conditions. If it was patchy at the start, it couldn’t be now nearly flat. The CMB would not look homogenous and isotropic.

... has been washed clean by an initial thermal equilbration — apokrisis

Heat is a measure of average energy over a region. Thermal equilibration is a statistical process. My remarks stand.

However no one wants to talk about the real combinatorial issues here — apokrisis

I'm perfectly happy to talk about combinatorics. Although I'm ignorant of the specific physics, I gather that the argument takes the form of modeling the number of particles in a region, and all the states they can be in, and figuring out how many possible configurations of all the particles there are.

But it doesn't matter. The number, whatever it is, is finite. As far as the mathematics, the situation is best modeled by assuming there are exactly two states, heads and tails. Many here are making detailed technical points of physics. But if you had a "zillion" possible states -- Tom objected to that earlier, a zillion is just whatever big finite number of states you like -- then you could code each state in binary and combine them by some rule and you'd map your entire state space into the set of all possible binary sequences.

Coin flips are exactly the same as fancy physics-y states for purposes of this discussion. Because you could express all the states in binary and have the same conversation.

Now Tom has claimed in one post that there are infinitely many states. THAT I do NOT believe. There must be finitely many states in order for this particular argument to go through at all. The argument is that there are infinitely many Hubble volumes, and each one can take on one of a bounded set of finitely many states. In other words there's some number S and that's the most states you can have. [In other words you can't have 1 state in Hubble 1, 2 states in Hubble 2, and so forth. There is a max number of states ANY region can have].

Otherwise this entire argument does not work. There are finitely many states. And when you're making statistical arguments, coin flips work just as well as huge numbers because you can always code any huge number in binary. The specific details of the physics and the calculation of the possible number of states is a huge distraction that's causing people to miss the fact that they are making statistical arguments. Statistics only tell you about populations, and never individuals.

no one wants to talk about the real combinatorial issues here. — apokrisis

We’re waiting for the quantum computer to come along.

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. — Marchesk

Yes, I think Vilenkin entertains similar fun scenarios, but frankly, not having followed the derivations, I am a little hesitant to commit to such specific predictions.

How can anybody assert that the state of some event outside our sphere 'is' in any particular state? Our definition sort of assumes a measurement taken from 'here', and by that definition, those distant events have no measurement and are in complete superposition. — noAxioms

Superposition states are states too (they are also called "mixed" states, as opposed to "pure" states). But I think I get your point: if we haven't been in contact with some remote region of the universe, then within that interval of time its wavefunction has been evolving independently from us, and there is no coherence between us and any one of its branches.

I think the cosmological principle allows such exceptions, but just says that the probability of us being that exception is sufficiently infinitesimal to preclude explanations that require us to be that exception. — noAxioms

This is simply false. In Inflationary cosmology, no assumptions about randomness, or all initial states being instantiated, or probability distributions, or typicality, or mediocrity is required. Inflation guarantees that this type of "exhaustive randomness" is in place.

Which is why I have been describing the initial conditions as ERGODIC from the beginning.

Superposition states are states too (they are also called "mixed" states, as opposed to "pure" states). But I think I get your point: if we haven't been in contact with some remote region of the universe, then within that interval of time its wavefunction has been evolving independently from us, and there is no coherence between us and any one of its branches. — SophistiCat

A type-1 alternate universe is just like a type-3 in that we might share a common portion of past history, but we can effectively no longer interact, ever. One is a past statement, and one is the future. The future makes it type-1, and that indeed is a mixed state. But for there to be a copy of Earth, we need a reasonably identical past, which would be a pure state since nothing can come from outside.

Neither are bounded by the Hubble-Sphere. The type-1 universe is bounded by the event horizon (which IS frame dependent, despite my expressed hesitancy in the prior post), but the Earth copy requires that pure quantum state which is bounded by the particle horizon.
The former is a ball about 31 Glyr in diameter (units in proper distance), but the latter is a frame-independent ball about 92 Glyr in diameter, beyond which all quantum states are pure from our standpoint. That means the nearest copy of us is only 92 Glyr away There are closer ones, but there is no coherence between us and them, so they don't really exist in a type-1 sense.

This is simply false. — tom

Could be. You need to reply to those who know this subject better than I. I've been a ball of disproven opinions on this point throughout this thread.
Comment on my QM thingy instead. I just stated that there is a copy of us quite nearby, to the point of giving a fairly specific figure for it.

They do not say that they are speaking precisely and formally in their books. It is only you that says that. The evidence points to the opposite being the case. The absence of equations is a big clue.

In any case, the books are not holy scripture and we are not in the helpless position of those trying to interpret holy scripture and work out what the Author intended. Either mathematical analysis supports a conclusion that there does not exist a single level 1 spacetime lacking a duplicate Earth, rather than the set of such spacetimes merely having measure zero, or it doesn't. If it does, you should be able to point to a rigorous proof of the former. So far you have not done so. — andrewk

Here's Tegmark being as explicit in one of his papers as in his book:

"In particular there are infinitely many other inhabited planets, including not just one, but infinitely many with people with the same appearance, name and memories as you. Indeed there are infinitely many other regions the size of our observable universe, where every possible cosmic history is played out. This is the Level I multiverse."

All relevant papers are available on arXiv.

EDIT:

For example the abstract of the paper you claimed to have read:

"A generic prediction of inflation is that the thermalized region we inhabit is spatially infinite. Thus, it contains an infinite number of regions of the same size as our observable universe, which we shall denote as O-regions. We argue that the number of possible histories which may take place inside of an O-region, from the time of recombination up to the present time, is finite. Hence, there are an infinite number of O-regions with identical histories up to the present, but which need not be identical in the future. Moreover, all histories which are not forbidden by conservation laws will occur in a finite fraction of all O-regions. The ensemble of O-regions is reminiscent of the ensemble of universes in the many-world picture of quantum mechanics. An important difference, however, is that other O-regions are unquestionably real.

Superposition states are states too (they are also called "mixed" states, as opposed to "pure" states). — SophistiCat

Actually it's the other way round. Superpositions are pure states, mixed states are statistical mixtures.

An article on arXiv is no more holy scripture than a pop-science book.

I asked you to link to a proof for your claim. Those papers contain no proofs.

The closest the Vilenkin paper comes is on p7 (2nd para of section IV) where it says:

All histories consistent with exact conservation laws will have non-vanishing probabilities and will occur in an infinite number of O-regions

Right there, in that sentence, Vilenkin asserts that E having a nonzero probability in a single trial entails that it is impossible for there to be an infinite sequence of trials in which E does not happen. That is, he simply assumes the conclusion that you assert. He does not prove it.

That's because he's writing informally. That becomes blindingly obvious in the paragraphs that follow, where he whimsically contemplates things like O-regions in which Elvis is still alive.

Max Tegmark says there are no infinities in the world.

He explicitly says that the infinities that come up in multiverse theory are breakdowns in our theory, and not actual infinities of universes.

http://blogs.discovermagazine.com/crux/2015/02/20/infinity-ruining-physics/

Also see my extended discussion here.

https://thephilosophyforum.com/discussion/comment/142382