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

is infinite qualitatively different from “really really big?” 187There are some rather strange aspects to everlasting (or “semi-eternal”) inflationthat may be resolved by this “truly eternal” sort of model.Consider the expectation value of the time T since the putative initial singularity seenby an observer in semi-eternal inflation. At each successive time there are exponentiallymore (but still finitely many) observers than at the previous time (if we assume thatobservers are born at some rate per unit physical volume). Thus, the expectation valueis infinite. But, of course, each observer was born at a finite time, so each observeris infinitely atypical, in the sense that the probability is zero of a randomly chosenobserver being born earlier than the chosen observer. In the truly eternal case, thisparadox goes away: there is no meaning, in the global sense, to any observer beingolder than any other. 9A similar paradox, sometimes called the “youngness paradox,” arises when weask about the time t since inflation ended found by a typical observer fulfilling some(non–age-related) criterion. Suppose, for example, that an “observer” must have someattribute A, and that A is correlated to t, but not in one-to-one relation with t (e.g., Amight be “is gravitationally bound” or “has density ρ”). Now, consider some bubbleand the observers fullfilling A. Most will have t A (the time that is “naturally” linkedwith attribute A), with some scatter. Consider all the observers existing at some Tand fulfilling A. Some will see a time t A , as expected. But because of the exponentialexpansion, exponentially more of them will have inflated until just prior to T, thenstopped inflating just in time to, by some unusual set of circumstances, attain attributeA in a time span shorter than t A . So almost all observers at a given T will see a timet ≪ t A , and the connection – based on theory – that A implies t A , will be broken. Thiscan be made fairly exact; for example, if we take A to be “contains a baryon,” then onlyan incredibly tiny fraction of all observers like us that exist at a given time T would seea microwave background temperature as low as we do Tegmark (2005).As far as I can ascertain, this paradox does not arise in the fully eternal model,because there is no meaning to T, or to saying that one T should contain exponentiallymore observers than another. But this leads to a strange dilemma: should the statisticalpredictions made in eternal inflation be continuous or discontinuous when the universeis changed from semi-eternal inflation to truly eternal inflation? If continuous, it seemsclear that no measure that explicitly makes use of the time T can make any sense. Ifdiscontinuous, this would imply a categorical distinction between the infinite and the“indefinitely large” that would be quite interesting.If we take the former view, it suggests searching for measures that correctly encodethe time translation invariance of the universe. One possibility would be to look fora global foliation of spacetime (both inside and outside of the bubbles) that accomplishesthis. Another choice would be to abandon the importance of a foliation andcompare regions at different “times,” as is apparently advocated by Linde (2007b), the9 Interestingly, this reasoning very closely parallels that used in the “doomsday” argument that it is unnatural toimagine that our civilization will go on much, much longer with exponential growth because this would meanthat we see extremely atypical values of the observable “people born before us.” Thus, I hereby propose a newresolution of the doomsday paradox: the universe is infinite at all times, and we must consider not just thepopulation of people in “our” civilization but all of the infinitely many others. Considering this whole set, thereis no sense in which the population is growing, and we are just as likely to live at any time as at any other.