On 5/2/2010 8:40 AM, Rex Allen wrote:
Returning to the thread:
On Sat, May 1, 2010 at 10:23 PM, Brent Meeker<meeke...@dslextreme.com> wrote:
On 5/1/2010 7:10 PM, Rex Allen wrote:
On Sat, May 1, 2010 at 10:01 PM, Brent Meeker<meeke...@dslextreme.com>
It's invalid simply because your conclusion depends taking the cardinality
to be the measure. The cardinality of infinite sets doesn't satisfy
Kolomogorov's axioms for a probability measure. For example one of the
axioms is: If A and B are disjoint then P(A) + P(B) = P(A union B). Let
the measure of the integers be 1. The let A be the evens and B be the odds.
You get 2=1. If you're going to talk about probabilities of infinite sets
you have to introduce some measure other than cardinality.
Isn't that exactly what I said here:
SO, I think we have zero information that we can use to base our
probability calculation on. Because of the counting issues introduced
by the infinity combined with the lack of pattern. There is no usable
No that's not the same. If you based the order on a die toss there would be
no pattern, but there would still be a measure even when the cardinality was
infinite. Your use of "information" is ambiguous. On the one hand you use
it to mean "no pattern" and then you assume that must be the same as "we
don't know the measure".
I claim vindication.
But having done so, what measure are you suggesting for this
particular example instead? For this particular example. Not for
general cases involving telephone surveys.
In my example, a die toss, measure is based on the symmetry of the die.
In that case it seems to me that we are ignoring the *actual* infinite
set of randomly generated results and only talking about the measure.
Why do you think there is an *actual* infinite set.
Effectively we're saying, "We have no useful information about the
random infinite set - because it's random...and infinite.
But we do have useful information. The die is symmetric, a fact we use
to hypothesize that the measure of each face is equal.
So let's go
back to the measure and ask what would we get if we generated another
number according to that definition."
So returning to my infinite row of numbered squares, let's say we take
the 1-squares and put them into a one-to-one correspondence with the
Now, let's put a sticker on each 1-square that says "A". And another
sticker on each not-1-square that says "B". Now, let's put them back
into an infinite row. What is the probability of hitting an B-square
with my randomly thrown dart? What is the probability of hitting a
It seems to me that we can't say anything about the actual infinite
set. We can only talk about various measures on it. Which is what
you said. But I think is still consistent with my original example.
I'd think that if you have an actual randomly generated infinite set,
then you can't draw any probabilistic conclusions about that infinite
set, even if you know how it was generated (e.g., dice rolling). You
can only draw conclusions about the measure that describes the
Right? Or wrong?
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?
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.
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?
You've just added a lot of words and hidden the problem of defining the
measure in the words "randomly arrange" and "select one at random". 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.
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