Re: Probabilities with infinite sets ( Was many (identical) universes)

From: Fred Chen <flipsu5.domain.name.hidden>
Date: Wed, 16 Feb 2000 22:37:08 -0800

Nail on the head, George!

GSLevy.domain.name.hidden wrote:

> In a message dated 02/14/2000 11:42:56 PM Pacific Standard Time,
> flipsu5.domain.name.hidden writes:
>
> >
> > Selwyn St Leger wrote:
> >
> > > > Date: Sun, 13 Feb 2000 12:58:59 -0800
> > > > From: Fred Chen <flipsu5.domain.name.hidden>
> > > > To: Saj Malhi <sajm.domain.name.hidden>
> > > > Cc: everything-list.domain.name.hidden
> > > > Subject: Re: many (identical) universes
> > >
> > > > Saj Malhi wrote:
> > >
> > > > > 2. The infinity of even numbers is as large as the infinity of all
> > real
> > > > > numbers. Only different classes / types of infinity have been shown
> to
> > be of
> > > > > different size.
> > > > >
> > > >
> > > > Hmmm...I would have thought the set of even numbers is much smaller
> than
> > the set
> > > > of reals, because between any two even numbers, you have infinitely
> more
> > reals.
> > > >
> > > > Fred
> > >
> > > Surely the even mumbers are a sub-set of the integers and thus of a
> > > lower order of infinity than the reals?
> > >
> > > Selwyn St Leger
> >
> > My previous intuition would have led me to think that there are fewer
> evens
> > than
> > integers, but more rigorously it seems possible to map evens to integers
> 1:1
> > by using
> > n->2n, and likewise for odds. However, it is difficult to imagine a way of
> > doing this
> > for rationals (or reals).
> >
> > I guess the issue for AUH would be: can we have meaningful probability with
> > infinities? Is the total number of evens 'half' the total number of
> integers,
> > so that
> > any randomly picked integer has a 50% chance of being even? Perhaps
> instead
> > of
> > discussing total number, we need number density.
> >
> > Fred
> >
> >
>
> Fred has put his finger on one of the most important issues involving MWI and
> quantum immortality. The meaning of probability in the context of the MW
> branching.
>
> Either the Shroedinger wave function splits into a finite number of branches
> or into an infinite number of branches.
>
> If the split is finite, calculating the probability of a given outcome x,
> given a starting situation S is trivial. Let's assume that the total number
> of branchesfrom S is N, of which n<N contain the given outcome. Then,
> assuming a **random** selection of any branch from the set of N branches,
> the probability of outcome x is given by:
>
> P{x} = n/N
>
> When the split is infinite, the probability calculation is not so obvious.
> Consider an infinite split of cardinality aleph null. Let's index each branch
> so that branch index = natural number. So we have branches "1", "2",
> "3",..... Now let us define the outcome x as (branch index = even). Let's
> calculate the probability of outcome x.
> Common sense indicates that total number N of branches (all natural numbers)
> = twice number n of branches with outcome x (i.e., even branches). Common
> sense however is misleading.
>
> Here is the key issue. Depending how the branches are sorted and the
> **random** selection is performed, the probability calculation will produce
> different results. If we sort the branches in the order "1", "2", "3",...
> then it seems that the even outcome x will occur 50% of the time. However,
> this particular sorting is entirely arbitrary. I could have sorted the
> branches as "1", "3", "2", "5", "7", "4", "9", "11", "6", "13", "15", "8",...
> that is having the density of the odd numbers = twice the density of the even
> numbers. With this arrangement all natural numbers are guaranteed to occur in
> the series. Yet a **random** sampling will generate a probability of picking
> an even number = 1/3. It seems then that the definition of probability in the
> context of infinite sets must include the specification of how these sets are
> ordered! In addition, it is not clear how a random sampling can be performed
> over the natural number, since we seems always to be biased toward the
> beginning of the scale (near zero) since for any finite number that is
> selected, we are guaranteed to have many more numbers above it than under it!
>
> In an attempt to find a way out of this dilemma, we could assume that the
> branching is really a continuum. So let us consider a line interval from 0 to
> 1 as a model. Consider an outcome x specified for every real real number.
> That is any real number either has property x or does not have it. This
> property could be for example the 2nd digit = 3. The sorting problem crops up
> again! I could order my reals any way I please with all the reals containing
> property x crowded in a tiny subinterval, say between 0 and 0.001. Any
> **random** sampling over interval (0,1) will be very unlikely to pick any
> real with the property 2nd digit = 3. So assuming a continuum does not solve
> the dilemma!.
>
> We know from experience that probability calculations in the MW do make
> sense!!! There must be something wrong with the math. Does any one of you
> know what? In this context the concept of limit seems to be suspect. I don't
> know if probability theory has been extended to cover infinite sets. Does any
> one of you know what the state of the art is in this subject?

Usually when the probabilities are calculated in QM, the physics is already known
beforehand, so that the object is written as a superposition of known possible
(outcome) states, and the normalized coefficients help give the probability. So we
see a lot of textbook examples of 1/sqrt(2) * psi_a + 1/sqrt(2) * psi_b, where the
possible outcomes a and b each have 50% probability known beforehand. In a sense,
this fixes the ordering of the branches.
Received on Wed Feb 16 2000 - 22:42:44 PST