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

From: <GSLevy.domain.name.hidden>
Date: Thu, 17 Feb 2000 01:05:21 EST

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?
Received on Wed Feb 16 2000 - 22:08:02 PST