Well, I cannot answer for Vilenkin or Tegmark, but I think they were speaking informally. — SophistiCat
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
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
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
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
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.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
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.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.
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.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.
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.I've quoted from Vilenkin's book. Nerither he not Tegmark were speaking informally. — tom
No it doesn't. You can't count your clones. Physics tells us that the cardinality of your clones is Aleph_0. — tom
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.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. — noAxioms
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.We need another assumption. the cosmological principle, which says in effect that there are no measure zero misbehaviors! — fishfry
So the cosmological principle says absolutely nothing about microphenomena such as whether a particular teensy-weensy arrangement of molecules like the Earth recurs. — andrewk
The cosmological principle states that each constant-time hypersurface of the universe ('this spacetime') is homogeneous and isotropic at the large scale. — andrewk
.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
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.but multiverse theory strikes me as suffering from the same type of problem — fishfry
The homogeneity part of the cosmological principle requires that mass-energy be uniformly distributed 'at the large scale'.Aren't you neglecting that the matter density must be uniform?
... has been washed clean by an initial thermal equilbration — apokrisis
However no one wants to talk about the real combinatorial issues here — apokrisis
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
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
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
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.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
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.This is simply false. — 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. — andrewk
Superposition states are states too (they are also called "mixed" states, as opposed to "pure" states). — SophistiCat
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.All histories consistent with exact conservation laws will have non-vanishing probabilities and will occur in an infinite number of O-regions
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