Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

190 cosmological intimations of infinityThus, it appears that if we ask, “Given some set of conditions A o , what shouldwe observe next?” the theory described by a finite-state system described by a timeindependentHamiltonian cannot give the right answer: most realizations of the conditionsA o are preceded by nothing like the “natural” history s i ···s o , nor will they furtherprecede anything like what we expect our universe to do. (For more discussion, seeDyson et al. (2002), which makes this old argument in detail in the modern context.)Now, this might seem an artificial setup, but as noted previously, a number of peoplehave suggested that if there is a fundamental positive cosmological constant (i.e., ifthe global minimum of potential energy for all fields is positive), then the universeshould have a finite number of states. It also seems plausible that, at least after a longperiod of evolution, the effective Hamiltonian would become time-independent. Fromthis standpoint, thermodynamics seems to lead to the conclusion that for us to observewhat we do cosmologically, the universe must have an infinite number of states, ratherthan any finite number, no matter how stupendously large. With an infinite number ofstates one can imagine also an indefinite increase in entropy, and that the universe neverachieves equilibrium, potentially Carroll and Chen (2004) (although not necessarilyBousso et al. (2006b), Page (2006) resolving the paradox.What are we to make of this? One possibility is that the frequentist reasoning thatleads to the result is simply incorrect, or that we should not be asking the questionswe are asking. But then we are forced to accept (as the Bayesian reasoning of, e.g.,Hartle and Srednicki (2007), would seem to imply) that there is no way to rule out thatwe are simply statistical fluctuations from equilibrium. A second possibility would bethat frequentist reasoning is correct in principle, but that the measure (or some otherelement of reasoning) that we are using is incorrect, and that if done correctly, therewould be a smooth transition from a huge number of states to an infinite set. 12 Third,the reasoning may indeed be telling us something profound: that the very coherence ofour experience means that the universe has infinite possibilities.8.6 ConclusionsIt seems inescapable that, as finite beings, we can never prove that the universe isphysically infinite: we cannot travel through infinite spaces or times or experience aninfinite number of states. Nonetheless, I have argued that in modern cosmology, wemay face the fascinating situation that the theories (particularly inflation) devised toexplain the finite observed region of the universe also naturally produce an infiniteuniverse, through a process called everlasting inflation. This can be the case even if the12 My current suspicion is that a better lens through which to view this paradox is the amount of information necessaryto select out the subensemble of systems satisfying A o . Imagine, for example, a closed and impenetrablebox with a book of Shakespearean sonnets in it. Now wait forever, keeping the box at constant temperature.The book will slowly disintegrate and eventually attain equilibrium, but if we wait long enough, the macrostatewill evolve back to the state of a book; this will disintegrate and eventually form another book (as well as allmanner of other objects.) Suppose we ask, “Given that we find a book, which one will it be?” Well, the amountof information we would have to specify (and entropy generate) to monitor the box for the vast eons of timenecessary for it to cycle into various books would be stupendous, and there would be no reasonable way toseparate the box from the nonequilibrium measuring environment.