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.

Brent

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