On 5/1/2010 12:31 PM, Rex Allen wrote:

On Thu, Apr 29, 2010 at 11:24 PM, Brent Meeker<meeke...@dslextreme.com> wrote:But if the universe arose from a quantum fluctuation, it would necessarily start with very low entropy since it would not be big enough to encode more than one or two bits at the Planck scale. If one universe can start that way then arbitrarily many can. So then it is no longer clear that the evolved brain is less probable than the Boltzmann brain.I asked Sean about the application of probability to the Boltzmann brain scenario on his blog:"So, in chapter 10 you rule out the possibility of the eternal recurrence scenario based on the low probability of an observer of our type (human) being surrounded by a non-equilibrium visible universe compared to the probability being a “boltzmann brain” human observer who pops into existence to find himself surrounded by chaos. As you say, in the eternal recurrence scenario there should be far far more of the later than of the former. Okay. So, my question: If the recurrences are really eternal, then shouldn’t there be infinitely many of BOTH types of observers? Countably infinite? And aren’t all countably infinite sets of equal size? So in an infinite amount of time we would accumulate one countably infinite set of our type of observer. And over that same amount of time we’d could also accumulate another countably infinite set of the “Boltzmann Brain” type of observer. The two sets would be of the same size…countably infinite. Right? So probabilistic reasoning wouldn’t apply here, would it? Especially not in a “block” universe where we don’t even have to wait for an infinite amount of time to pass."AND, here was his reply:Sean Says: January 27th, 2010 at 9:49 am Rex, this is certainly a good problem, related to the “measure” issue that cosmologists are always talking about. Yes, in an eternal universe there are countably infinite numbers of “ordinary” observers and freak (thermal-fluctuation) observers. But the frequency of the latter — the average number in any particular length of time — is much larger. We generally assume that this is enough to calculate probabilities, although it’s hardly an airtight principle.

`Seems like a good answer to me. Suppose there were infinitely many`

`rolls of a die (which frequentist statisticians assume all the time).`

`The fact that the number of "1"s would be countably infinite and the`

`number of "not-1"s would be countably infinite would change the fact`

`that the "not-1"s are five times more probable.`

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