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.)