- Contemporary messages sorted: [ by date ] [ by thread ] [ by subject ] [ by author ] [ by messages with attachments ]

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

Date: Wed, 16 Feb 2000 22:37:08 -0800

Nail on the head, George!

GSLevy.domain.name.hidden wrote:

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

*
This archive was generated by hypermail 2.3.0
: Fri Feb 16 2018 - 13:20:06 PST
*