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In this paper, Sean Carroll basically argues that the ΛCDM/'flat lambda' cosmological model has to be wrong since it implies that we're probably Boltzmann brains (and therefore shouldn't trust our evidence in favour of ΛCDM in the first place). I'm curious though - assuming that ΛCDM is accurate, should we expect infinitely many Boltzmann brains?

Carroll claims that ΛCDM means that our universe asymptotically approaches a 'de Sitter phase' (p2,p12). He says that, in principle, de Sitter space can last forever (p3) and that it forms "an eternal thermal system" (p12). On the horizon of that de Sitter space we'd get quantum fluctuations ("dynamical processes in which entropy decreases, resulting from stochastic dynamics in time-dependent states" - p12-13) which would include, for instance, entire human brains. Because of this, Carroll says, we should expect there to be far more Boltzmann observers than ordinary observers.

Carroll doesn't say this explicitly, but doesn't it also mean that there should be infinitely many Boltzmann observers? If the system lasts forever, and Boltzmann observers come into existence with some non-zero, finite probability, shouldn't P(at least N Boltzmann observers exist) approach 1 as t->∞, for any finite number N? This means that the expected number of Boltzmann observers throughout the universe is unboundedly high, so we have to assign the total number an infinite cardinality, right?

I'd also like to know if anyone else says this in print (or argues against it). Carroll cites Albrecht & Sorbo, but they only seem to mention Boltzmann brains in passing, and don't seem to make any explicit mention of how many there should be. Does anyone else talk about the issue, and potentially imply that there are infinitely many Boltzmann brains?

Also, when talking about Boltzmann brains arising from thermodynamic or quantum fluctuations, is it fair to say that these fluctuations can produce any possible arrangement of matter / any possible physical phenomenon? Should we not only expect infinitely many Boltzmann brains but also infinitely many complete humans, infinitely many planets, infinitely many star systems, and infinitely many of all these things lasting for any amount of time (and having lasting conscious experience)?

(Apologies if these are stupid questions - I have zero background in physics. I'm actually interested in this for a philosophy paper, as it turns out that the number of conscious beings in the universe is potentially really important for whether certain ethical theories work or not.)

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  • $\begingroup$ Your question is well stated. It is not at all your fault that it is based on a bunch of nonsense on top of nonsense. There is no way to address this in a comment, but the biggest flaw is assuming that anything can be infinite. It cannot. Nothing infinite exists in the universe by the very definition of existence (sorry, out of scope here). Furthermore, the many-words interpretation is a nonsense on its own. Plus our universe is not de Sitter and will never be. All this is a brain exercise in abstract logic (short of the illogical infinity) , but has nothing to do with reality or the universe. $\endgroup$
    – safesphere
    Feb 6, 2018 at 8:34
  • $\begingroup$ @safesphere I hope this isn't too far off the beam, but could you maybe link or cite whatever text (preferably, given my rather low education level, the one getting the point across with a minimum of formalism) might effectively claim that no infinity can exist? (The Wiki on it includes a link to a mathematical proof by Cantor which seems to claim that at least a mathematical infinity can exist.) $\endgroup$
    – Edouard
    Feb 15, 2020 at 19:55
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    $\begingroup$ @Edouard In physics, infinity cannot be measured, so it is not a physical observable, period. In math, there is no point on the number line for infinity. It does not exist, period. Yet for each word of reason you hear a hundred words of nonsense presented with authority as “the ultimate truth”. They tell you it is not as simple and invent increasingly complex concepts, but only to dig a deeper logical hole for themselves. You can explore this matter further by searching the web for “Wildberger infinity” as a start. This guy has many accessible posts and videos and is an expert on the subject. $\endgroup$
    – safesphere
    Feb 15, 2020 at 22:56
  • $\begingroup$ I get safesphere's point about infinity not being a physical observable (as per the "in-" part of its name), but neither is its non-existence (as per the "non-" part of its own name). The closest approach infinity makes to being observable may, given the abundance of spherical objects in astrophysics, be that possibly-infinite divisibility which is suggested by pi. At least until (and if) that limit to the divisibility of space and time (which seems to be the holy grail of quantum physics) is found, infinity and eternality may remain as irresoluble as the Boltzmann brain possibility. $\endgroup$
    – Edouard
    Feb 21, 2020 at 20:55

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As far I can tell, open universes like the de Sitter spacetime would indeed get an infinite number of Boltzmann brains (and Boltzmann planets, Boltzmann galaxies, and Boltzmann worlds - but these are exponentially rarer as they get larger, so the typical anomalous observer will be a Boltzmann brain). I have not seen many mentions in the literature that they would be unboundedly many, but clearly most authors seem to realise it.

The most common approach appears to be to consider the density of observers, although this is often done in a way that ignores the limiting state (I think this is what is being done here, for example). This paper calculates numbers but only deals with their ratios. Basically, it is easier to deal with the case of the number of Boltzmann brains "now" than integrated across the future, especially since it is bad enough to produce an interesting problem. And right "now" presumably the number of Boltzmann brains are finite if the universe-at-large is finite.

(I don't think the question is stupid - sometimes it is relevant to spell out the assumptions carefully; I find many of the physics papers gloss over a lot of assumptions here.)

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    $\begingroup$ The present is a temporal focus of the subjective perception. There is no objective definition of the present moment of time. No physical law defines "now" or how old the universe is "now" objectively aside from you being a subjective observer. Hypothetically, if you die and are reincarnated, your next life may be at a completely different age of the universe, like double or a half of what it is today. In other words, the 4D universe is static. The phenomenon of "now" as "space moving in time" is purely subjective and is different for different observers who "live" in different epohs. $\endgroup$
    – safesphere
    Feb 6, 2018 at 7:51

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