On 5/3/2010 7:41 PM, Rex Allen wrote:
Probablistic statements are always about measure.  What you write above is
true, but it is also true if you substitute "finite" for "infinite".  It's
just that when you have a finite set or you are generating a potentially
inifinite set, then the cardinality and the relative rate of generation
provides a canonical measure.  But it's not the only one and not even
necessarily the right one depending on the problem - did you read the paper
I sent?

So I did read your handout on probability.  And I'm *still* not an
expert on the subject.  SO.  Epic fail on your part.



So, given eternal recurrence, there are an infinite number of Rexs.
And an infinite number of not-Rexs.  Let's pair the Rexs off in a
one-to-one correspondence with the not-Rexs.  Then, let's go down the
list and put an "A" sticker on the Rexs.  And a "B" sticker on the
not-Rexs.  Then lets randomly arrange them in an infinitely long row
and select one at random.  What's the probability of selecting a Rex?
What's the probability of selecting an "A" sticker?

I suppose your intent is to assign equal measure to each position on
the list so, for any finite subsection of the list the measure of As
and Bs will be equal.

If that was my intent, what would your response be?



But, now returning to the Boltzmann brain problem, Carroll says:

"This version of the multiverse will feature both isolated Boltzmann
brains lurking in the empty de Sitter regions, and ordinary observers
found in the aftermath of the low-entropy beginnings of the baby
universes. Indeed, there should be an infinite number of both types.
So which infinity wins?   The kinds of fluctuations that create freak
observers in an equilibrium background are certainly rare, but the
kinds of fluctuations that create baby universes are also very rare."

Well, since these are physically existing infinities of the same size,
then neither infinity wins.
Carroll is hoping that future advances is physics will tell us the relative
rate of creation of freak observers as compared to normal observers.  This
will then provide a measure that is independent of whether the number is
finite or infinite; just like we can say the probability of not-1 on die
throw is five times as likely as a 1 throw independent of any assumption
about the number of throws.

Okay so let's say that Carroll is correct.  And after an infinite
amount of time we end up with an infinite number of Boltzmann Brains
(BB) and Normal Brains (NB).

Now, even at the end of infinity,

From a logical contradiction everything follows.

physics still hasn't advanced to the
point that we can infer from theory what the relative rate of creation
of each brain type is.

BUT...(somehow) we have access to a record of what actually
happened...an infinite set of time-indexed data that shows a (BB) plus
a time-stamp for each boltzmann brain that was created and a (NB) plus
a time-stamp for each normal brain.

Now, what can we say about this infinite set?  Can we reconstruct a
probability distribution from it and have any confidence that this
measure accurately reflects the true  nature of the actual physical
processes that explain the distribution of the two different kinds of
brains?

In other words, if (unknown to us) in reality what controlled the
relative rate of creation was the equivalent of the random results of
a fair 6-sided die being rolled - where a "1" meant a Boltzmann brain
would be created while a "not-1" meant that a Normal brain would be
created - could we recover the fact of that 16.67% Boltzmann brain
creation rate using just the data in our infinite data set?
Sure. The (self-contradictory) assumption that we have reached the end of an infinite process is just a diversion. The events are time ordered so we can take arbitrarily large sample sequences and infer relative rates. You may raise Hume's objection to inductive inference, but that is quite independent of whether the sequence in infinite or finite.

Brent

Or would the random nature of the generation process plus the infinite
nature of the data set result in us being unable to recover that
information with high confidence?

If at the end of time we have the same number of boltzmann brains and
of normal brains...then I'm not sure what difference it makes to talk
about the measure that generated them.  There's the same number of
each type.  There's a 1-to-1 mapping between the subset of BB's and
the subset of NB's in our infinite dataset.  Isn't there?


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