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

200 infinity and the nostalgia of the stars9.3 Neverending TerritoryThese results imply that space is very close to a flat, or Euclidean, geometry. A spatiallyflat universe is, of course, spatially infinite. If this is really the case, and if we takeseriously a “strong cosmological principle,” we conclude that we live in a limitlessdistribution of matter and radiation extending to infinity in all directions with a highdegree of isotropy and homogeneity on large scales. The universe is characterizedeverywhere by the same basic ingredients and physical laws that we find in our cosmicsurroundings. Therefore, the only changes from one region to another in the boundlessdepth of space would be due to the random local outcomes of the same basic processes.Because the spatial extension is infinite, one should also conclude that any systemoccurring with a nonzero probability, no matter how improbable, must be realized againand again an infinite number of times. The very fact that we see a given object, say,the Matterhorn, means that that particular object has nonzero probability to be formed.Therefore, the Matterhorn must have infinite identical copies existing somewhere inthe universe. The formation process of DNA-based life forms, even if exceedinglyunlikely, should be expected to take place indefinitely often in the infinite universe.If we consider a large enough portion of space, the same exact history that life hasundergone on our planet, its evolution to complex organisms up to the emergenceof consciousness, should be found to occur identically in some remote planets. 17 If,in addition, one is prepared to think that conscious beings, such as ourselves, arecompletely defined by the physical systems supporting them, then deeper paradoxesarise (Tegmark 2003). Somewhere, very far away, right now there would be an infinityof beings indistinguishable from yourself, reading exactly these same words about theinfinity of the universe. . . .The degree of weirdness one can fantasize about by playing with probabilities inan infinite universe is rather arbitrary. This situation reflects unsolved problems indefining a meaningful measure, that is, how to compute what is common and rarewithin an infinite set (Bousso 2006). Recently, a discussion has developed around onesuch extreme case, the so-called Boltzmann brain paradox. We know that quantumfield fluctuations can materialize particles with a very small, but nonzero, probability.If instead of a single particle we now want, say, the quantum formation of a fullcarbon atom complete with its twelve nucleons and electrons perfectly arranged, thenthe probability will be vastly smaller – but still nonzero. In principle, we also have atiny chance of a sudden quantum appearance of rocks, Matterhorns, complex systems,cells, and so on. “Boltzmann’s brains” are conscious observers – naked brains in emptyspace! – popping out of quantum vacuum for a very short time, perhaps endowed witha full array of virtual memories of a fictitious past personal history. Some recent calculationsin connection with inflationary models (Dyson, Kleban, and Susskind 2002;17 The situation is further exacerbated in models invoking a “perfect cosmological principle,” which postulatesthat the average properties are maintained not only in spatial extension but also in time. In such a universeanything that is happening around you right now not only is happening elsewhere infinitely often, but it hasalways happened and will happen endlessly, in the past and in the future. Such a scenario was popular in theearly 1960s through the Steady-State model of Fred Hoyle, Tommy Gold, and Hermann Bondi. The modelwas then discarded under observational evidence against it, most notably after the discovery of the CMB in1965 by Arno Penzias and Robert Wilson.